## 3.7 Inverse Functions

Learning objectives.

In this section, you will:

- Verify inverse functions.
- Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
- Find or evaluate the inverse of a function.
- Use the graph of a one-to-one function to graph its inverse function on the same axes.

A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.

If some physical machines can run in two directions, we might ask whether some of the function “machines” we have been studying can also run backwards. Figure 1 provides a visual representation of this question. In this section, we will consider the reverse nature of functions.

## Verifying That Two Functions Are Inverse Functions

Betty is traveling to Milan for a fashion show and wants to know what the temperature will be. She is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, Betty wants to convert 75 degrees Fahrenheit to degrees Celsius using the formula

and substitutes 75 for F F to calculate

Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, Betty gets the week’s weather forecast from Figure 2 for Milan, and wants to convert all of the temperatures to degrees Fahrenheit.

At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for F F after substituting a value for C . C . For example, to convert 26 degrees Celsius, she could write

After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.

The formula for which Betty is searching corresponds to the idea of an inverse function , which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.

Given a function f ( x ) , f ( x ) , we represent its inverse as f − 1 ( x ) , f − 1 ( x ) , read as “ f “ f inverse of x . ” x . ” The raised −1 −1 is part of the notation. It is not an exponent; it does not imply a power of −1 −1 . In other words, f − 1 ( x ) f − 1 ( x ) does not mean 1 f ( x ) 1 f ( x ) because 1 f ( x ) 1 f ( x ) is the reciprocal of f f and not the inverse.

The “exponent-like” notation comes from an analogy between function composition and multiplication: just as a − 1 a = 1 a − 1 a = 1 (1 is the identity element for multiplication) for any nonzero number a , a , so f − 1 ∘ f f − 1 ∘ f equals the identity function, that is,

This holds for all x x in the domain of f . f . Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal , some functions do not have inverses.

Given a function f ( x ) , f ( x ) , we can verify whether some other function g ( x ) g ( x ) is the inverse of f ( x ) f ( x ) by checking if both g ( f ( x ) ) = x g ( f ( x ) ) = x and f ( g ( x ) ) = x f ( g ( x ) ) = x are true.

For example, y = 4 x y = 4 x and y = 1 4 x y = 1 4 x are inverse functions.

A few coordinate pairs from the graph of the function y = 4 x y = 4 x are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function y = 1 4 x y = 1 4 x are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

## Inverse Function

For any one-to-one function f ( x ) = y , f ( x ) = y , a function f − 1 ( x ) f − 1 ( x ) is an inverse function of f f if f − 1 ( y ) = x . f − 1 ( y ) = x . This can also be written as f − 1 ( f ( x ) ) = x f − 1 ( f ( x ) ) = x for all x x in the domain of f . f . It also follows that f ( f − 1 ( x ) ) = x f ( f − 1 ( x ) ) = x for all x x in the domain of f − 1 f − 1 if f − 1 f − 1 is the inverse of f . f .

The notation f − 1 f − 1 is read “ f “ f inverse.” Like any other function, we can use any variable name as the input for f − 1 , f − 1 , so we will often write f − 1 ( x ) , f − 1 ( x ) , which we read as “ f “ f inverse of x . ” x . ” Keep in mind that

and not all functions have inverses.

## Identifying an Inverse Function for a Given Input-Output Pair

If for a particular one-to-one function f ( 2 ) = 4 f ( 2 ) = 4 and f ( 5 ) = 12 , f ( 5 ) = 12 , what are the corresponding input and output values for the inverse function?

The inverse function reverses the input and output quantities, so if

Alternatively, if we want to name the inverse function g , g , then g ( 4 ) = 2 g ( 4 ) = 2 and g ( 12 ) = 5. g ( 12 ) = 5.

Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed. See Table 1 .

Given that h − 1 ( 6 ) = 2 , h − 1 ( 6 ) = 2 , what are the corresponding input and output values of the original function h ? h ?

Given two functions f ( x ) f ( x ) and g ( x ) , g ( x ) , test whether the functions are inverses of each other.

- Determine whether f ( g ( x ) ) = x f ( g ( x ) ) = x or g ( f ( x ) ) = x . g ( f ( x ) ) = x .
- If either statement is true, then both are true, and g = f − 1 g = f − 1 and f = g − 1 . f = g − 1 . If either statement is false, then both are false, and g ≠ f − 1 g ≠ f − 1 and f ≠ g − 1 . f ≠ g − 1 .

## Testing Inverse Relationships Algebraically

If f ( x ) = 1 x + 2 f ( x ) = 1 x + 2 and g ( x ) = 1 x − 2 , g ( x ) = 1 x − 2 , is g = f − 1 ? g = f − 1 ?

We must also verify the other formula.

Notice the inverse operations are in reverse order of the operations from the original function.

If f ( x ) = x 3 − 4 f ( x ) = x 3 − 4 and g ( x ) = x + 4 3 , g ( x ) = x + 4 3 , is g = f − 1 ? g = f − 1 ?

## Determining Inverse Relationships for Power Functions

If f ( x ) = x 3 f ( x ) = x 3 (the cube function) and g ( x ) = 1 3 x , g ( x ) = 1 3 x , is g = f − 1 ? g = f − 1 ?

No, the functions are not inverses.

The correct inverse to the cube is, of course, the cube root x 3 = x 1 3 , x 3 = x 1 3 , that is, the one-third is an exponent, not a multiplier.

If f ( x ) = ( x − 1 ) 3 and g ( x ) = x 3 + 1 , f ( x ) = ( x − 1 ) 3 and g ( x ) = x 3 + 1 , is g = f − 1 ? g = f − 1 ?

## Finding Domain and Range of Inverse Functions

The outputs of the function f f are the inputs to f − 1 , f − 1 , so the range of f f is also the domain of f − 1 . f − 1 . Likewise, because the inputs to f f are the outputs of f − 1 , f − 1 , the domain of f f is the range of f − 1 . f − 1 . We can visualize the situation as in Figure 3 .

When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of f ( x ) = x f ( x ) = x is f − 1 ( x ) = x 2 , f − 1 ( x ) = x 2 , because a square “undoes” a square root; but the square is only the inverse of the square root on the domain [ 0 , ∞ ) , [ 0 , ∞ ) , since that is the range of f ( x ) = x . f ( x ) = x .

We can look at this problem from the other side, starting with the square (toolkit quadratic) function f ( x ) = x 2 . f ( x ) = x 2 . If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and –3. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the “inverse” is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.

In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function f ( x ) = x 2 f ( x ) = x 2 with its domain limited to [ 0 , ∞ ) , [ 0 , ∞ ) , which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).

If f ( x ) = ( x − 1 ) 2 f ( x ) = ( x − 1 ) 2 on [ 1 , ∞ ) , [ 1 , ∞ ) , then the inverse function is f − 1 ( x ) = x + 1. f − 1 ( x ) = x + 1.

- The domain of f f = range of f − 1 f − 1 = [ 1 , ∞ ) . [ 1 , ∞ ) .
- The domain of f − 1 f − 1 = range of f f = [ 0 , ∞ ) . [ 0 , ∞ ) .

Is it possible for a function to have more than one inverse?

No. If two supposedly different functions, say, g g and h , h , both meet the definition of being inverses of another function f , f , then you can prove that g = h . g = h . We have just seen that some functions only have inverses if we restrict the domain of the original function. In these cases, there may be more than one way to restrict the domain, leading to different inverses. However, on any one domain, the original function still has only one unique inverse.

## Domain and Range of Inverse Functions

The range of a function f ( x ) f ( x ) is the domain of the inverse function f − 1 ( x ) . f − 1 ( x ) .

The domain of f ( x ) f ( x ) is the range of f − 1 ( x ) . f − 1 ( x ) .

Given a function, find the domain and range of its inverse.

- If the function is one-to-one, write the range of the original function as the domain of the inverse, and write the domain of the original function as the range of the inverse.
- If the domain of the original function needs to be restricted to make it one-to-one, then this restricted domain becomes the range of the inverse function.

## Finding the Inverses of Toolkit Functions

Identify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on which each function is one-to-one, if any. The toolkit functions are reviewed in Table 2 . We restrict the domain in such a fashion that the function assumes all y -values exactly once.

The constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no inverse.

The absolute value function can be restricted to the domain [ 0 , ∞ ) , [ 0 , ∞ ) , where it is equal to the identity function.

The reciprocal-squared function can be restricted to the domain ( 0 , ∞ ) . ( 0 , ∞ ) .

We can see that these functions (if unrestricted) are not one-to-one by looking at their graphs, shown in Figure 4 . They both would fail the horizontal line test. However, if a function is restricted to a certain domain so that it passes the horizontal line test, then in that restricted domain, it can have an inverse.

The domain of function f f is ( 1 , ∞ ) ( 1 , ∞ ) and the range of function f f is ( −∞ , −2 ) . ( −∞ , −2 ) . Find the domain and range of the inverse function.

## Finding and Evaluating Inverse Functions

Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.

## Inverting Tabular Functions

Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.

Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.

## Interpreting the Inverse of a Tabular Function

A function f ( t ) f ( t ) is given in Table 3 , showing distance in miles that a car has traveled in t t minutes. Find and interpret f − 1 ( 70 ) . f − 1 ( 70 ) .

The inverse function takes an output of f f and returns an input for f . f . So in the expression f − 1 ( 70 ) , f − 1 ( 70 ) , 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function f , f , 90 minutes, so f − 1 ( 70 ) = 90. f − 1 ( 70 ) = 90. The interpretation of this is that, to drive 70 miles, it took 90 minutes.

Alternatively, recall that the definition of the inverse was that if f ( a ) = b , f ( a ) = b , then f − 1 ( b ) = a . f − 1 ( b ) = a . By this definition, if we are given f − 1 ( 70 ) = a , f − 1 ( 70 ) = a , then we are looking for a value a a so that f ( a ) = 70. f ( a ) = 70. In this case, we are looking for a t t so that f ( t ) = 70 , f ( t ) = 70 , which is when t = 90. t = 90.

Using Table 4 , find and interpret ⓐ f ( 60 ) , f ( 60 ) , and ⓑ f − 1 ( 60 ) . f − 1 ( 60 ) .

## Evaluating the Inverse of a Function, Given a Graph of the Original Function

We saw in Functions and Function Notation that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.

Given the graph of a function, evaluate its inverse at specific points.

- Find the desired input on the y -axis of the given graph.
- Read the inverse function’s output from the x -axis of the given graph.

## Evaluating a Function and Its Inverse from a Graph at Specific Points

A function g ( x ) g ( x ) is given in Figure 5 . Find g ( 3 ) g ( 3 ) and g − 1 ( 3 ) . g − 1 ( 3 ) .

To evaluate g ( 3 ) , g ( 3 ) , we find 3 on the x -axis and find the corresponding output value on the y -axis. The point ( 3 , 1 ) ( 3 , 1 ) tells us that g ( 3 ) = 1. g ( 3 ) = 1.

To evaluate g − 1 ( 3 ) , g − 1 ( 3 ) , recall that by definition g − 1 ( 3 ) g − 1 ( 3 ) means the value of x for which g ( x ) = 3. g ( x ) = 3. By looking for the output value 3 on the vertical axis, we find the point ( 5 , 3 ) ( 5 , 3 ) on the graph, which means g ( 5 ) = 3 , g ( 5 ) = 3 , so by definition, g − 1 ( 3 ) = 5. g − 1 ( 3 ) = 5. See Figure 6 .

Using the graph in Figure 5 , ⓐ find g − 1 ( 1 ) , g − 1 ( 1 ) , and ⓑ estimate g − 1 ( 4 ) . g − 1 ( 4 ) .

## Finding Inverses of Functions Represented by Formulas

Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula—for example, y y as a function of x — x — we can often find the inverse function by solving to obtain x x as a function of y . y .

Given a function represented by a formula, find the inverse.

- Make sure f f is a one-to-one function.
- Solve for x . x .
- Interchange x x and y . y .
- Replace y y with f - 1 ( x ) f - 1 ( x ) . (Variables may be different in different cases, but the principle is the same.)

## Inverting the Fahrenheit-to-Celsius Function

Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.

By solving in general, we have uncovered the inverse function. If

In this case, we introduced a function h h to represent the conversion because the input and output variables are descriptive, and writing C − 1 C − 1 could get confusing.

Solve for x x in terms of y y given y = 1 3 ( x − 5 ) . y = 1 3 ( x − 5 ) .

## Solving to Find an Inverse Function

Find the inverse of the function f ( x ) = 2 x − 3 + 4. f ( x ) = 2 x − 3 + 4.

So f − 1 ( y ) = 2 y − 4 + 3 f − 1 ( y ) = 2 y − 4 + 3 or f − 1 ( x ) = 2 x − 4 + 3. f − 1 ( x ) = 2 x − 4 + 3.

The domain and range of f f exclude the values 3 and 4, respectively. f f and f − 1 f − 1 are equal at two points but are not the same function, as we can see by creating Table 5 .

## Solving to Find an Inverse with Radicals

Find the inverse of the function f ( x ) = 2 + x − 4 . f ( x ) = 2 + x − 4 .

So f − 1 ( x ) = ( x − 2 ) 2 + 4. f − 1 ( x ) = ( x − 2 ) 2 + 4.

The domain of f f is [ 4 , ∞ ) . [ 4 , ∞ ) . Notice that the range of f f is [ 2 , ∞ ) , [ 2 , ∞ ) , so this means that the domain of the inverse function f − 1 f − 1 is also [ 2 , ∞ ) . [ 2 , ∞ ) .

The formula we found for f − 1 ( x ) f − 1 ( x ) looks like it would be valid for all real x . x . However, f − 1 f − 1 itself must have an inverse (namely, f f ) so we have to restrict the domain of f − 1 f − 1 to [ 2 , ∞ ) [ 2 , ∞ ) in order to make f − 1 f − 1 a one-to-one function. This domain of f − 1 f − 1 is exactly the range of f . f .

What is the inverse of the function f ( x ) = 2 − x ? f ( x ) = 2 − x ? State the domains of both the function and the inverse function.

## Finding Inverse Functions and Their Graphs

Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function f ( x ) = x 2 f ( x ) = x 2 restricted to the domain [ 0 , ∞ ) , [ 0 , ∞ ) , on which this function is one-to-one, and graph it as in Figure 7 .

Restricting the domain to [ 0 , ∞ ) [ 0 , ∞ ) makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.

We already know that the inverse of the toolkit quadratic function is the square root function, that is, f − 1 ( x ) = x . f − 1 ( x ) = x . What happens if we graph both f f and f − 1 f − 1 on the same set of axes, using the x - x - axis for the input to both f and f − 1 ? f and f − 1 ?

We notice a distinct relationship: The graph of f − 1 ( x ) f − 1 ( x ) is the graph of f ( x ) f ( x ) reflected about the diagonal line y = x , y = x , which we will call the identity line, shown in Figure 8 .

This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.

## Finding the Inverse of a Function Using Reflection about the Identity Line

Given the graph of f ( x ) f ( x ) in Figure 9 , sketch a graph of f − 1 ( x ) . f − 1 ( x ) .

This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of ( 0 , ∞ ) ( 0 , ∞ ) and range of ( − ∞ , ∞ ) , ( − ∞ , ∞ ) , so the inverse will have a domain of ( − ∞ , ∞ ) ( − ∞ , ∞ ) and range of ( 0 , ∞ ) . ( 0 , ∞ ) .

If we reflect this graph over the line y = x , y = x , the point ( 1 , 0 ) ( 1 , 0 ) reflects to ( 0 , 1 ) ( 0 , 1 ) and the point ( 4 , 2 ) ( 4 , 2 ) reflects to ( 2 , 4 ) . ( 2 , 4 ) . Sketching the inverse on the same axes as the original graph gives Figure 10 .

Draw graphs of the functions f f and f − 1 f − 1 from Example 8 .

Is there any function that is equal to its own inverse?

Yes. If f = f − 1 , f = f − 1 , then f ( f ( x ) ) = x , f ( f ( x ) ) = x , and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because

Any function f ( x ) = c − x , f ( x ) = c − x , where c c is a constant, is also equal to its own inverse.

Access these online resources for additional instruction and practice with inverse functions.

- Inverse Functions
- One-to-one Functions
- Inverse Function Values Using Graph
- Restricting the Domain and Finding the Inverse

## 3.7 Section Exercises

Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

Why do we restrict the domain of the function f ( x ) = x 2 f ( x ) = x 2 to find the function’s inverse?

Can a function be its own inverse? Explain.

Are one-to-one functions either always increasing or always decreasing? Why or why not?

How do you find the inverse of a function algebraically?

Show that the function f ( x ) = a − x f ( x ) = a − x is its own inverse for all real numbers a . a .

For the following exercises, find f − 1 ( x ) f − 1 ( x ) for each function.

f ( x ) = x + 3 f ( x ) = x + 3

f ( x ) = x + 5 f ( x ) = x + 5

f ( x ) = 2 − x f ( x ) = 2 − x

f ( x ) = 3 − x f ( x ) = 3 − x

f ( x ) = x x + 2 f ( x ) = x x + 2

f ( x ) = 2 x + 3 5 x + 4 f ( x ) = 2 x + 3 5 x + 4

For the following exercises, find a domain on which each function f f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f f restricted to that domain.

f ( x ) = ( x + 7 ) 2 f ( x ) = ( x + 7 ) 2

f ( x ) = ( x − 6 ) 2 f ( x ) = ( x − 6 ) 2

f ( x ) = x 2 − 5 f ( x ) = x 2 − 5

Given f ( x ) = x 2 + x f ( x ) = x 2 + x and g ( x ) = 2 x 1 − x : g ( x ) = 2 x 1 − x :

- ⓐ Find f ( g ( x ) ) f ( g ( x ) ) and g ( f ( x ) ) . g ( f ( x ) ) .
- ⓑ What does the answer tell us about the relationship between f ( x ) f ( x ) and g ( x ) ? g ( x ) ?

For the following exercises, use function composition to verify that f ( x ) f ( x ) and g ( x ) g ( x ) are inverse functions.

f ( x ) = x − 1 3 f ( x ) = x − 1 3 and g ( x ) = x 3 + 1 g ( x ) = x 3 + 1

f ( x ) = − 3 x + 5 f ( x ) = − 3 x + 5 and g ( x ) = x − 5 − 3 g ( x ) = x − 5 − 3

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

f ( x ) = x f ( x ) = x

f ( x ) = 3 x + 1 3 f ( x ) = 3 x + 1 3

f ( x ) = −5 x + 1 f ( x ) = −5 x + 1

f ( x ) = x 3 − 27 f ( x ) = x 3 − 27

For the following exercises, determine whether the graph represents a one-to-one function.

For the following exercises, use the graph of f f shown in Figure 11 .

Find f ( 0 ) . f ( 0 ) .

Solve f ( x ) = 0. f ( x ) = 0.

Find f − 1 ( 0 ) . f − 1 ( 0 ) .

Solve f − 1 ( x ) = 0. f − 1 ( x ) = 0.

For the following exercises, use the graph of the one-to-one function shown in Figure 12 .

Sketch the graph of f − 1 . f − 1 .

Find f ( 6 ) and f − 1 ( 2 ) . f ( 6 ) and f − 1 ( 2 ) .

If the complete graph of f f is shown, find the domain of f . f .

If the complete graph of f f is shown, find the range of f . f .

For the following exercises, evaluate or solve, assuming that the function f f is one-to-one.

If f ( 6 ) = 7 , f ( 6 ) = 7 , find f − 1 ( 7 ) . f − 1 ( 7 ) .

If f ( 3 ) = 2 , f ( 3 ) = 2 , find f − 1 ( 2 ) . f − 1 ( 2 ) .

If f − 1 ( − 4 ) = − 8 , f − 1 ( − 4 ) = − 8 , find f ( − 8 ) . f ( − 8 ) .

If f − 1 ( − 2 ) = − 1 , f − 1 ( − 2 ) = − 1 , find f ( − 1 ) . f ( − 1 ) .

For the following exercises, use the values listed in Table 6 to evaluate or solve.

Find f ( 1 ) . f ( 1 ) .

Solve f ( x ) = 3. f ( x ) = 3.

Solve f − 1 ( x ) = 7. f − 1 ( x ) = 7.

Use the tabular representation of f f in Table 7 to create a table for f − 1 ( x ) . f − 1 ( x ) .

For the following exercises, find the inverse function. Then, graph the function and its inverse.

f ( x ) = 3 x − 2 f ( x ) = 3 x − 2

f ( x ) = x 3 − 1 f ( x ) = x 3 − 1

Find the inverse function of f ( x ) = 1 x − 1 . f ( x ) = 1 x − 1 . Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

## Real-World Applications

To convert from x x degrees Celsius to y y degrees Fahrenheit, we use the formula f ( x ) = 9 5 x + 32. f ( x ) = 9 5 x + 32. Find the inverse function, if it exists, and explain its meaning.

The circumference C C of a circle is a function of its radius given by C ( r ) = 2 π r . C ( r ) = 2 π r . Express the radius of a circle as a function of its circumference. Call this function r ( C ) . r ( C ) . Find r ( 36 π ) r ( 36 π ) and interpret its meaning.

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, t , t , in hours given by d ( t ) = 50 t . d ( t ) = 50 t . Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function t ( d ) . t ( d ) . Find t ( 180 ) t ( 180 ) and interpret its meaning.

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## Included in version 2.61 released 8/27/2021:

- Fixed: Preferences page for Kuta Works
- Improved: Security and stability with updated networking libraries

## Included in version 2.60 released 8/19/2021:

- Improved: Site licenses check expiration date more frequently
- Fixed: Combinatoric Word Problems - Values could overflow
- Fixed: Properties of Exponents - Answer might not be fully simplified
- Fixed: Properties of Logs - Possible to have duplicate choices
- Fixed: Radical Equations - Option to mix radicals and rational exponents had no effect

## Included in version 2.52 released 6/14/2019:

- New: Scramble questions by directions
- New: Scramble all questions in assignment
- New: Consolidate question sets with identical options
- New: Regroup question sets by directions
- New: Kuta Works - Create a two semester course
- New: Kuta Works - Clone assignments from a previous course into a new one
- Fixed: Presentation View - Arrow keys could change zoom at start/end of assignment
- Fixed: Tight Layout - Directions could be cramped for no reason
- Fixed: Inverse Trig - Angles limited for tan( x )
- Fixed: Inverse Trig - Errors in UI logic

## Included in version 2.50 released 4/12/2019:

- New: Kuta Works - Option to hide answers and results from students until after due date
- New: Kuta Works - Option to control how long choices are hidden
- Improved: Options windows appear at a better initial size
- Improved: Window controls shown at their native size and spacing
- Improved: Network proxy configuration window
- Improved: License activation process
- Fixed: Presentation View - Answers could be cut off
- Fixed: Some window controls cut off on Windows 10 with display scaling
- Fixed: Spelling errors
- Fixed: Kuta Works - Course list could display more than just your active courses
- Fixed: Graphing Rational Functions - Open holes in graph could appear as filled holes

## Included in version 2.42 released 12/11/2018:

- Fixed: Certain symbols incorrect if saved on Mac and loaded on Windows or vice versa
- Fixed: General Sequences - Recursive formula sometimes wrong
- Fixed: Properties of Logarithms - Avoid questions with reducible roots

## Included in version 2.41 released 9/26/2018:

- Fixed: Archived courses in Kuta Works are now hidden
- Fixed: Generating or regenerating questions could cause a crash

## Included in version 2.40 released 8/8/2018:

- New: Integrated with Kuta Works . Now post assignments online.

## Included in version 2.25 released 4/27/2018:

- Improved: Generate questions more quickly
- Improved: Measurement arrows reach correct endpoints
- Improved: Absolute Value Equations - Better decimal numbers
- Improved: Absolute Value Inequalities - Better decimal numbers
- Improved: Compound Inequalities - Better decimal numbers
- Improved: Multi-Step Equations - Better decimal numbers and more predictable special cases
- Improved: Multi-Step Inequalities - Better decimal numbers
- Improved: Order of Operations - Better decimal numbers
- Fixed: Trig units for piecewise functions and ordered pairs
- Fixed: Graphing Rational Functions - Prevent equivalent choices
- Fixed: Law of Sines - Program could freeze
- Fixed: Transformations - Prevent equivalent choices

## Included in version 2.18 released 7/31/2017:

- Fixed: Open circles on graphs were filled in when displayed as answer in red
- Fixed: Typo in sample custom question
- Fixed: Transformations - Program could freeze
- Fixed: Systems of Quadratic Equations - Program could freeze

## Included in version 2.17 released 4/27/2017:

- Improved: [Windows] "Export to clipboard as bitmap" renders image with better quality
- Fixed: Probability with Combinations and Permutations - Could ask about impossible lottery ball

## Included in version 2.16.20 released 11/16/2016:

- Fixed: Solving Quadratic Equations by Taking Square Roots - Option to "Allow fractions" not working as expected

## Included in version 2.16 released 9/16/2016:

- Fixed: Numbers with many significant digits could round incorrectly
- Fixed: Customized question containing an error could not be modified
- Fixed: Customized question containing an error could cause a crash
- Fixed: Feedback tool detects Windows 10 properly
- Fixed: Mouse cursor when drag-dropping a question
- Fixed: Keyboard shortcut for changing order of question sets in assignment

## Included in version 2.15 released 7/14/2016:

- Improved: Reorder question sets by dragging and dropping
- Improved: Move individual questions within a set by dragging and dropping
- Improved: Presentation View - Commands now on a movable, dockable toolbar
- Improved: Added option for large toolbar icons
- Improved: New icons
- Fixed: Certain options could result in double parenthesis or parenthesis around a negative sign
- Fixed: Arrow over vector variable name not red when appropriate
- Fixed: Solving Systems of Equations by Graphing - Answer could be displayed twice
- Fixed: Equations of Parabolas - Certain givens indicate wrong sign

## Included in version 2.11 released 5/13/2016:

- Improved: [Windows] "Export to clipboard as bitmap" pastes into Microsoft Word at appropriate DPI
- Fixed: Keyboard shortcut for adding space to question set

## Included in version 2.10 released 5/11/2016:

- New: Preference for including "None of the above" as a choice
- New: One column layout
- New: Added support for Infinite Precalculus
- New: Easily add piecewise functions of graphs in custom questions: Example: piecewise([2x-3] if [x<5], [x-1] if [x >= 5])
- New: Easily add functions with restricted domains to graphs in custom questions Example: function(x/2, x<0)
- Improved: More efficient layout of choices
- Improved: [Mac] Added option for legal paper in page setup
- Fixed: Spelling and grammatical errors
- Fixed: Graphing Quadratic Functions - Prevent equivalent choices
- Fixed: Binomial Theorem - Certain options didn't allow for many questions
- Fixed: Right Triangle Trig - Couldn't select just one ratio except sine
- Fixed: Right Triangle Trig - Answer sometimes given in problem
- Fixed: Law of Cosines - Certain SAS cases could freeze program
- Note: Older versions will not be able to open assignments saved with this version

## Included in version 2.06 released 8/3/2015:

- Fixed: Crash when saving if previously used folder no longer exists

## Included in version 2.05 released 7/20/2015:

- New Topic: Sample Spaces and The Fundamental Counting Principle
- New Topic: Independent and Dependent Events
- New Topic: Independent and Dependent Events Word Problems
- New Topic: Mutually Exclusive Events
- New Topic: Mutually Exclusive Events Word Problems
- Fixed: Classifying Conics - Checkboxes for conic types had no effect
- Fixed: Presentation View - Displayed wrong directions in 3-up mode
- Fixed: Presentation View - Lines too long after creating new problems
- Fixed: [Windows] Uncommon error when using the network

## Included in version 2.04.40 released 3/31/2015:

- Fixed: Activation data unrecognized under special circumstances
- Fixed: [Windows] Help files broken by previous release
- Fixed: Order of Operations - Radio buttons reversed

## Included in version 2.04.20 released 3/23/2015:

- Improved: [Windows] "Export to clipboard as bitmap" uses a transparent background

## Included in version 2.04 released 3/17/2015:

- New Topic: Permutations
- New Topic: Combinations
- New Topic: Permutations vs Combinations
- New Topic: Probability with Permutations and Combinations
- New Topic: Converting Degrees and Degrees-Minutes-Seconds
- [factorial(n)]
- [perm(n, r)]
- [comb(n, r)]
- Improved: Better support for proxies
- Improved: Support for HiDPI Mac Retina displays
- Improved: Print options window - Some settings are now persistent
- Improved: Presentation View - Use arrow keys to navigate between questions
- Improved: Question set updates free response/multiple-choice when all questions have changed
- Improved: [Mac] Activation also checks /Library/Preferences/ks-config.txt for serial numbers
- Fixed: [Windows] Crash when printing with certain printers
- Fixed: Crash when loading an assignment with over 80 custom questions
- Fixed: Custom question answer was never choice A
- Fixed: Paper size set before printing
- Fixed: Cramer's Rule - Fixed directions
- Fixed: Systems of Three Equations - Restored the option to "only solve for x"
- Fixed: Center of circle wasn't appearing

## Included in version 2.03 released 9/19/2014:

- Fixed: Preference for question layout mode not correctly loaded
- Fixed: Issues with automatic spacing and highlighting questions when using tight layout
- Fixed: In Presentation View, graphing problems display choices more compactly
- Fixed: Printing blank pages on HP LaserJet 1018, 1020, and 1022 among others
- Fixed: Logs and Exponents as Inverses - problem with slider

## Included in version 2.02 released 8/5/2014:

- New: Software uses system proxy when necessary
- Improved: Activation window has taskbar entry
- Improved: Remembers the last directory used when saving and opening files
- Fixed: [Windows only] Possible crash when loading an assignment with a lot of custom questions
- Fixed: [Mac only] Application stops responding to hover events
- Fixed: Activation from before v2.0 sometimes not recognized
- Fixed: Export to clipboard sometimes cuts off choices B and D
- Fixed: Help contents could become unusable
- Fixed: Word wrap when modifying a question
- Fixed: Memory leak

## Included in version 2.01 released 7/28/2014:

- Fixed: Could crash when opening an assignment from a different program

## Included in version 2.00 released 7/23/2014:

- New: Print to PDF
- New: Filter for topic list
- New: Zoom while viewing on screen
- New: "More like these" button in presentation mode
- New: Show where punch holes will go on the page
- New: Print only an answer sheet
- New: When printing, page range is enabled
- New: Page elements are outlined when hovered
- New: Double-click on directions to change them
- New: When merging assignments, options to put similar topics together or put them end-to-end
- New: Highlight and go to the questions in a question set (menu command or shift-click)
- New: Site licenses can be activated per-machine if run by the administrator
- Ctrl-Click on a question
- Shift-Click on a question
- Ctrl-Shift-Click on a question
- Shift-Click in a empty area of a page
- Ctrl-Shift-Click in a empty area of a page
- Shift-Click on a question set
- Ctrl-Shift-Click on a question set
- Improved: Assignment files are drastically smaller
- Improved: Sidebar is dockable on either side, can be floated, and position reset
- Improved: Auto-spacing doesn't leave dead space at bottom of each page
- Improved: Preview when editing directions
- Improved: Keyboard shortcuts work no matter where the focus is
- Improved: Scale number of questions window more user-friendly
- Fixed: Minor bugs in seven topics

## Included in version 1.56 released 8/14/2013:

- Fixed: Graphing Exponential Functions - Correct choice used lighter shade of blue
- Fixed: Radical Equations could be missing a solution
- Fixed: Graphs of Polynomial Functions - Inflection points sometimes listed as local extrema
- Fixed: Properties of Exponents - Occasionally answer wasn't fully simplified
- Fixed: Dividing Polynomials - Occasionally polynomial was evenly divisible when it shouldn't have been
- Fixed: Graphing Rational Functions - Minor UI issue
- Fixed: Geometric Sequences - Given information sometimes allowed for two possible common ratios
- Fixed: Dividing Polynomials could freeze
- Fixed: Systems of Equations by Elimination - "No solution" occasionally should have been "Infinite solutions"

## Included in version 1.55 released 12/11/2012:

- New: Graphs can be added to custom questions
- Improved: Graphing and Graph Paper utility more powerful and easier to use
- Improved: Support for loading files from Infinite Calculus
- Improved: Faster save/load
- Fixed: Answer for Factoring Quadratic Expressions sometimes incorrect
- Fixed: Choices for Evaluating Functions with a variable operand could contain wrong variable
- Fixed: Choices for Function Operations with a variable operand could contain wrong variable
- Fixed: Function Operations, functions when composed with self could provide the wrong answer
- Fixed: Custom questions with an illegal expression could freeze the program
- Fixed: Certain families of functions graphed incorrectly

## Included in version 1.53 released 9/11/2012:

- New: Preference for notation for greatest integer function
- New: Added maximize/restore button to Presentation View
- New: Graphing Absolute Value Equations has options for including coefficients
- Improved: Help files
- Improved: Scroll bars
- Improved: User interface
- Improved: When choices make a question too tall for a page, some choices are removed
- Improved: Algebraic simplification routines are now more efficient
- Improved: Better graphs for: Graphing Absolute Value Equations, Graphing Linear Equations, Graphing Exponential Functions, Graphing Quadratic Functions, Graphing Linear Inequalities, Graphing Systems of Linear Inequalities
- Improved: Better number lines for: Multi-Step Inequalities, Compound Inequalities, Absolute Value Inequalities
- Fixed: Wrapping to full-page and half-page could be too wide
- Fixed: Able to graph certain families of functions
- Fixed: Issue involving loading & regenerating an assignment and answers being hidden
- Fixed: Program could ask about changing directions too much
- Fixed: Graphs for Compound Inequalities could omit interval
- Fixed: Double & Half Angle Identities: Button to show more examples did not work
- Fixed: Writing Linear Equations: Answer can't be line given in question

## Included in version 1.52 released 5/29/2012:

- Improved: More professional radical signs
- Improved: Graphs of discontinuous functions have breaks and open / closed holes
- Improved: Even less likely to crash when generating questions
- Fixed: Word wrapping could skip blank lines, not wrap where appropriate
- Fixed: Wrapping to full page in a custom question was too wide
- Fixed: Products of powers of e (like 2 e ³) no longer display with a multiplication dot
- Fixed: Most diagrams appeared incorrectly in print preview
- Fixed: Random error message on save or load
- Fixed: Graphing Rational Functions: Open holes in graph could appear as filled holes

## Included in version 1.51.02 released on 4/9/2012:

- Fixed: Minimum Windows version was set incorrectly

## Included in version 1.51 released 4/5/2012:

- New: Added greatest integer function / floor function to custom questions Example: Find [int(2x)] when [x=3/5]
- New: Added piecewise functions to custom questions Example: [eval(f,x)] = [piecewise [x] if [x < 0], [xx] if [x >= 0]]
- Improved: Faster! Optimized rendering of questions to screen
- Improved: Faster! Optimized graphing routines
- Improved: Improved graphing capabilities
- Improved: Diagrams drawn more smoothly on screen
- Fixed: Rewriting Logarithms - Numeric problems sometimes had bad values
- Fixed: Evaluating Logarithms - Problems sometimes had bad values
- Fixed: Graphs could omit holes
- Fixed: Graphs of constant functions or those involving e and π could be incorrect
- Fixed: Minor indentation issue in custom questions
- Fixed: Horizontal asymptotes could be drawn beyond a graph's area

## Included in version 1.50 released 3/15/2012:

- New: Presentation View window is resizeable
- New: Presentation View has option to automatically hide the answers when a new question is displayed
- New: High-level filter to prevent questions from containing an illegal expression
- Improved: Function notation can now be edited
- Improved: Faster! Optimized the simplification of mathematical expressions
- Improved: Faster! Improved undo/redo algorithm
- Improved: Smaller executable size
- Improved: "Current Question Sets" list easier to use
- Improved: Options windows resize more smoothly
- Fixed: Equation of Parabolas could freeze
- Fixed: Geometric Series questions could list "Illegal Expression" as a choice
- Fixed: Graphing Exponential Functions could freeze
- Fixed: Systems of Equations Word Problems could freeze
- Fixed: Punctuation in some word problems
- Fixed: Certain expressions in a custom question would cause the software to crash
- Fixed: Expressions like root × term would not print a multiplication dot
- Note: Beginning with this version, Windows XP SP3 is the minimum required version of Windows

## Included in version 1.45 released 4/12/2011:

- New: Assignments from this program can be opened and modified by our other programs (Assignments saved with v1.45 or greater can be opened by other programs v1.45 or greater)
- New: Student data fields (name, date, period) can be renamed
- New: Additional student data field available
- New: Add & Continue is available when modifying an existing question set
- New: Link to this details page when a software update is available
- New: Product serial numbers can be placed in config.txt to facilitate enterprise installations
- Improved: Better backwards- and future-compatibility
- Fixed: Assignments with quotes in the title don't prevent Save As
- Fixed: Quotients could be sometimes be simplified incorrectly
- Fixed: Choices for dilations sometimes inappropriate
- Fixed: Dilations rounding x- and y-coordinates
- Fixed: Function Operations could have "Illegal Expression" as an answer or as a choice
- Fixed: Multi-Step Equations: Program could freeze
- Fixed: Dividing Radical Expressions: Program could freeze
- Fixed: Add & Continue respects change in problem type
- Fixed: Powers of i are correctly displayed

## Included in version 1.42 released on 6/28/2010:

- New: Automatically checks for updates
- New: List topics by index order or suggested order

## Included in version 1.40 released 2/8/2010:

- New Preference: 'Spacious' or 'tight' layout of questions on page
- Improved: Question sets with one question are more intelligently spaced
- Improved: Graphs with small ranges now look better
- Improved: Options screen now resizeable
- Fixed: Directions occasionally orphaned at bottom of page
- Fixed: Correct numbers changed because zeros at the end of a number were deleted (50 --> 5)

## Included in version 1.08.20 released 5/22/2009:

- Changed: Minor change to license agreement so that renewals extend the termination date instead of beginning a new term.

## Included in version 1.08 released 4/10/2009:

- New: Possible to change directions for a set without regenerating questions
- Improved: Changing directions cascades changes to appropriate neighbors
- Correction: Answers to Properties of Exponents could have wrong sign
- Correction: Inequalities were sometimes solved incorrectly
- Fixed: Software sometimes failed to renumber questions in assignment

## Included in version 1.07 released 1/6/2009:

- New: Trig. equations that don't require factoring
- Correction: Function operations / linear combinations answers sometimes wrong
- Fixed: Software crashes when starting if closed when side panel is minimized

## Included in version 1.06 released 12/4/2008:

- Improved: Isolate up to four questions on screen at once
- Improved: Support for computers with multiple processors
- Improved: User interface: Toolbar + Preferences
- Fixed: Obscure error involving selecting a question and scaling the assignment

## Included in version 1.03 released 10/1/2008:

- New: Export individual questions as bitmaps
- Improved: Custom questions wrap to whole/half page
- Improved: Insert special symbols and math in custom questions
- Improved: Insert 'blanks' into math text
- Improved: Line thickness preference also now controls darkness of grid lines in graphs
- Correction: Writing logs in terms of others sometimes created duplicate questions
- Fixed: Internet calls would sometimes report wrong error

## Included in version 1.02 released 9/18/2008:

- Correction: Question type was not saving all of its options
- Correction: Equations of Hyperbolas description was sometimes wrong
- Correction: Geometric Series directions was sometimes wrong
- Correction: Binomial Theorem type could cause the program to freeze
- Correction: Quadratic/Linear systems sometimes had the wrong answer
- Correction: Rational Exponents answers now correct
- Fixed: Internet calls could freeze program

## Solver Title

## Generating PDF...

- Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form Mean, Median & Mode
- Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
- Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
- Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
- Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
- Linear Algebra Matrices Vectors
- Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
- Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
- Physics Mechanics
- Chemistry Chemical Reactions Chemical Properties
- Finance Simple Interest Compound Interest Present Value Future Value
- Economics Point of Diminishing Return
- Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
- Pre Algebra
- Pre Calculus
- Given Points
- Given Slope & Point
- Slope Intercept Form
- Start Point
- Parallel Lines
- Perpendicular
- Perpendicular Lines
- Perpendicular Slope
- Is a Function
- Domain & Range
- Slope & Intercepts
- Periodicity
- Domain of Inverse
- Critical Points
- Inflection Points
- Monotone Intervals
- Extreme Points
- Global Extreme Points
- Absolute Extreme
- Turning Points
- End Behavior
- Average Rate of Change
- Piecewise Functions
- Discontinuity
- Values Table
- Compositions
- Arithmetics
- Circumference
- Eccentricity
- Conic Inequalities
- Transformation
- Linear Algebra
- Trigonometry
- Conversions

## Most Used Actions

Number line.

- inverse\:y=\frac{x^2+x+1}{x}
- inverse\:f(x)=x^3
- inverse\:f(x)=\ln (x-5)
- inverse\:f(x)=\frac{1}{x^2}
- inverse\:y=\frac{x}{x^2-6x+8}
- inverse\:f(x)=\sqrt{x+3}
- inverse\:f(x)=\cos(2x+5)
- inverse\:f(x)=\sin(3x)
- How do you calculate the inverse of a function?
- To calculate the inverse of a function, swap the x and y variables then solve for y in terms of x.
- What are the 3 methods for finding the inverse of a function?
- There are 3 methods for finding the inverse of a function: algebraic method, graphical method, and numerical method.
- What is the inverse of a function?
- The inverse of a function f is a function f^(-1) such that, for all x in the domain of f, f^(-1)(f(x)) = x. Similarly, for all y in the domain of f^(-1), f(f^(-1)(y)) = y
- Can you always find the inverse of a function?
- Not every function has an inverse. A function can only have an inverse if it is one-to-one so that no two elements in the domain are matched to the same element in the range. A non-one-to-one function is not invertible.

function-inverse-calculator

- Functions A function basically relates an input to an output, there’s an input, a relationship and an output. For every input...

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## Inverse Functions Worksheet

Students will practice work with inverse functions including identifying the inverse functions , graphing inverses and more.

## Example Questions

Directions: Find the inverse of each function

## Challenge Problems. Part II

- Inverse Function
- Relations and Functions -- everything you might want to know.
- Domain and Range
- Functions and Relations in Math

## Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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## Algebra 2 (FL B.E.S.T.)

Course: algebra 2 (fl b.e.s.t.) > unit 7.

- Unit test Inverse functions

## MATH 375. Elementary Numerical Analysis (with Python)

1. root finding by interval halving (bisection) ¶.

References:

Section 1.1 The Bisection Method of Numerical Analysis by Sauer

Section 2.1 The Bisection Method of Numerical Analysis by Burden&Faires

(See the References .)

## 1.1. Introduction ¶

One of the most basic tasks in numerical computing is finding the roots (or “zeros”) of a function — solving the equation \(f(x) = 0\) where \(f:\mathbb{R} \to \mathbb{R}\) is a continuous function from and to the real numbers. As with many topics in this course, there are multiple methods that work, and we will often start with the simplest and then seek improvement in several directions:

reliability or robustness — how good it is at avoiding problems in hard cases, such as division by zero.

accuracy and guarantees about accuracy like estimates of how large the error can be — since in most cases, the result cannot be computed exactly.

speed or cost — often measure by minimizing the amount of arithemtic involved, or the number of times that a function must be evaluated.

Example 1: Solve \(x = \cos x\) . This is a simple equation for which there is no exact formula for a solution, but we can easily ensure that there is a solution, and moreover, a unique one. It is convenient to put the equation into “zero-finding” form \(f(x) = 0\) , by defining

Also, note that \(|\cos x| \leq 1\) , so a solution to the original equation must have \(|x| \leq 1\) . So we will start graphing the function on the interval \([a, b] = [-1, 1]\) .

Aside: This is our first use of two Python packages that some of you might not have seen before: Numpy and Matplotlib . If you want to learn more about them, see for example the Python Review sections on Python Variables, Lists, Tuples, and Numpy arrays and Graphing with Matplotlib

Or for now, just learn from the examples here.

This shows that the zero lies between 0.5 and 0.75, so zoom in:

And we could repeat, geting an approximation of any desired accuracy.

However this has two weaknesses: it is very inefficient (the function is evaluated about fifty times at each step in order to draw the graph), and it requires lots of human intervention.

To get a procedure that can be efficiently implemented in Python (or another programming language of your choice), we extract one key idea here: finding an interval in which the function changes sign, and then repeatedly find a smaller such interval within it. The simplest way to do this is to repeatedly divide an interval known to contain the root in half and check which half has the sign change in it.

Graphically, let us start again with interval \([a, b] = [-1, 1]\) , but this time focus on three points of interest: the two ends and the midpoint, where the interval will be bisected:

Aside on Numpy’s math functions: note on line 3 above that the function cos from Numpy (full name numpy.cos ) can be evaluated simultaneously on a list of numbers; the version math.cos from module math can only handle one number at a time. This is one reason why we will avoid math in favor of numpy .

\(f(a)\) and \(f(c)\) have the same sign, while \(f(c)\) and \(f(b)\) have opposite signs, so the root is in \([c, b]\) ; update the a, b, c values and plot again:

Again \(f(c)\) and \(f(b)\) have opposite signs, so the root is in \([c, b]\) , and …

This time \(f(a)\) and \(f(c)\) have opposite sign, so the root is at left, in \([a, c]\) :

## 1.2. A first algorithm for the bisection method ¶

Now it is time to dispense with the graphs, and describe the procedure in mathematical terms:

if \(f(a)\) and \(f(c)\) have opposite signs, the root is in interval \([a, c]\) , which becomes the new version of interval \([a, b]\) .

otherwise, \(f(c)\) and \(f(b)\) have opposite signs, so the root is in interval \([c, b]\)

## 1.2.1. Pseudo-code for describing algorithms ¶

As a useful bridge from the mathematical desciption of an algorithm with words and formulas to actual executable code, these notes will often describe algorithms in pseudo-code — a mix of words and mathematical formulas with notation that somewhat resembles code in a language like Python.

This is also preferable to going straight to code in a particular language (such as Python) because it makes it easier if, later, you wish to implement algorithms in a different programming language.

Note well one feature of the pseudo-code used here: assignment is denoted with a left arrow:

\(x \leftarrow a\)

is the instruction to cause the value of variable x to become the current value of a.

This is to distinguish from

which is a comparison : the true-or-false assertion that the two quantities already have the same value.

Unfortunately however, Python (like most programming languages) does not use this notation: instead assignment is done with x = a so that asserting equality needs a differnt notation: this is done with x == a ; note well that double equal sign!

Also, the pseudo-code marks the end of blocks like if , for and while with the lines end if , end for , end while and so on. Many programming languages do something like this (or just use end for all blocks) but Python does not: instead it uses only the end of indentation as the indication that a block is finished.

With those notational issues out of the way, the key step in the bisection strategy is the update of the interval:

\(\displaystyle c \leftarrow \frac{a + b}{2}\) if \(f(a) f(c) < 0\) then: \(\quad\) \(b \leftarrow c\) else: \(\quad\) \(a \leftarrow c\) end if

This needs to be repeated a finite number of times, and the simplest way is to specify the number of iterations. (We will consider more refined methods soon.)

Get an initial interval \([a, b]\) with a sign-change: \(f(a) f(b) < 0\) .

Choose \(N\) , the number of iterations.

for i from 1 to N: \(\quad\) \(\displaystyle c \leftarrow \frac{a + b}{2}\) \(\quad\) if \(f(a) f(c) < 0\) then: \(\quad\) \(\quad\) \(b \leftarrow c\) \(\quad\) else: \(\quad\) \(\quad\) \(a \leftarrow c\) \(\quad\) end if end for

The approximate root is the final value of \(c\) .

A Python version of the iteration is not a lot different:

(If you wish to review for loops in Python, see the Python Review section on Iteration with for )

## 1.2.2. Exercise 1 ¶

Create a Python function bisection1 which implements the first algorithm for bisection abive, which performd a fixed number \(N\) of iterations; the usage should be: root = bisection1(f, a, b, N)

Test it with the above example: \(f(x) = x - \cos x = 0\) , \([a, b] = [-1, 1]\)

(If you wish to review the defining and use of functions in Python, see the Python Review section on Defining and Using Python Functions )

## 1.3. Error bounds, and a more refined algorithm ¶

The above method of iteration for a fixed number of times is simple, but usually not what is wanted in practice. Instead, a better goal is to get an approximation with a guaranteed maximum possible error: a result consisting of an approximation \(\tilde{r}\) to the exact root \(r\) and also a bound \(E_{max}\) on the maximum possible error; a guarantee that \(|r - \tilde{r}| \leq E_{max}\) . To put it another way, a guarantee that the root \(r\) lies in the interval \([\tilde{r} - E_{max}, \tilde{r} + E_{max}]\) .

In the above example, each iteration gives a new interval \([a, b]\) guaranteed to contain the root, and its midpoint \(c = (a+b)/2\) is with a distance \((b-a)/2\) of any point in that interval, so at each iteration, we can have:

\(\tilde{r}\) is the current value of \(c = (a+b)/2\)

\(E_{max} = (b-a)/2\)

## 1.4. Error tolerances and stopping conditions ¶

The above algorthm can passively state an error bound, but it is better to be able to solve to a desired degree of accuracy; for example, if we want a result “accurate to three decimal places”, we can specify \(E_{max} \leq 0.5 \times 10^{-3}\) .

So our next goal is to actively set an accuracy target or error tolerance \(E_{tol}\) and keep iterating until it is met. This can be achieved with a while loop; here is a suitable algorithm:

Input function \(f\) , interval endpoints \(a\) and \(b\) , and an error tolerance \(E_{tol}\)

Evaluate \(E_{max} = (b-a)/2\)

while \(E_{max} > E_{tol}\) : \(\quad c \leftarrow (a+b)/2\) \(\quad\) if \(f(a) f(c) < 0\) then: \(\quad\quad b \leftarrow c\) \(\quad\) else: \(\quad\quad a \leftarrow c\) \(\quad\) end if \(\quad E_{max} \leftarrow (b-a)/2\) end while

Output \(\tilde{r} = c\) as the approximate root and \(E_{max}\) as a bound on its absolute error.

(If you wish to review while loops, see the Python Review section on Iteration with while )

## 1.4.1. Exercise 2 ¶

Create a Python function implementing this better algorithm, with usage root = bisection2(f, a, b, E_tol)

Test it with the above example: \(f(x) = x - \cos x\) , \([a, b] = [-1, 1]\) , this time accurate to within \(10^{-4}\) .

Use the fact that there is a solution in the interval \((-1, 1)\) .

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## Inverse of 2×2 Matrix with Examples

Inverse of 2×2 matrix is the matrix obtained by dividing the adjoint of the matrix by the determinant of the matrix. The two methods to find the inverse of 2×2 matrix is by using inverse formula and by using elementary operations.

In this article, we will explore how to find the inverse of 2×2 matrix along with both the methods and basics of the inverse of matrix. We will also solve some examples of the inverse of 2×2 matrix. Let’s start our learning on the topic “How to Find the Inverse of 2×2 Matrix?”.

Table of Content

## What is Inverse of Matrix?

How to find the inverse of 2×2 matrix, inverse of 2×2 matrix by inverse formula, inverse of 2×2 matrix by elementary operations, solved examples on inverse of 2×2 matrix.

The inverse of matrix is referred to as the matrix which, when multiplied by its original matrix gives identity matrix. The inverse of matrix A is represented as A -1 . If you have a square matrix A, its inverse matrix A -1 , such that when A is multiplied by A -1 , the result is the identity matrix (I).

Mathematically this relationships is represented as:

A × A -1 = A -1 × A = I

The inverse of a matrix can only be determined for a square and non-singular matrix (i.e., determinant of matrix is non-zero).

The two ways to find the inverse of 2×2 matrix is:

The steps to find inverse of 2×2 matrix by inverse formula is listed.

Step 1: First, find the determinant of 2×2 matrix. Step 2: Then, find the adjoint of the 2×2 matrix. Step 3: Put the values of determinant and adjoint of the 2×2 matrix in formula: A -1 = adj(A) / |A| Step 4: Then, we get the inverse formula of 2×2 matrix.

Inverse Formula for Inverse of 2×2 Matrix

The inverse formula for Inverse of 2×2 matrix A = [Tex]\begin{bmatrix} a &b\\ c & d \end{bmatrix}[/Tex] is given by:

A -1 = [Tex]\bold{\frac{1}{ad – bc} \begin{bmatrix} d &-b\\ -c & a \end{bmatrix}}[/Tex]

The steps to find inverse of 2×2 matrix by elementary operations is listed below.

Step 1: First, write the matrix as A = IA where I is the identity matrix of order 2×2. Step 2: Then, perform row elementary operation or column elementary operation until we get identity matrix on the LHS. Step 3: When identity matrix I is achieved by performing the row or column operation then we get I = BA. Step 4: Then, B represents the inverse of 2×2 matrix A.

Read More About

- Elementary Operation
- Determinant
- Inverse of Matrix by Elementary Operation
- Inverse of 3×3 Matrix

Example 1: Find the inverse of matrix B = [Tex]\begin{bmatrix} 10 & 5\\ 7 & 3 \end{bmatrix}[/Tex] using inverse formula

B = [Tex]\begin{bmatrix} 10 & 5\\ 7 & 3 \end{bmatrix} [/Tex] The inverse of 2×2 matrix formula is given by: B -1 = [Tex]\bold{\frac{1}{ad – bc} \begin{bmatrix} d &-b\\ -c & a \end{bmatrix}}[/Tex] ⇒ B -1 = [Tex]\frac{1}{(10\times 3) – (7\times 5)} \begin{bmatrix} 3 &-5\\ -7 & 10 \end{bmatrix}[/Tex] ⇒ B -1 = [Tex]\frac{1}{ 30 – 35} \begin{bmatrix} 3 &-5\\ -7 & 10 \end{bmatrix} [/Tex] ⇒ B -1 = [Tex]\frac{1}{-5} \begin{bmatrix} 3 &-5\\ -7 & 10 \end{bmatrix} [/Tex]

Example 2: Find the inverse of matrix C = [Tex]\begin{bmatrix} 30 & 8\\ 7 & 1 \end{bmatrix}[/Tex] using inverse formula.

C = [Tex]\begin{bmatrix} 30 & 8\\ 7 & 1 \end{bmatrix} [/Tex] The inverse of 2×2 matrix formula is given by: C -1 = [Tex] \bold{\frac{1}{ad – bc} \begin{bmatrix} d &-b\\ -c & a \end{bmatrix}}[/Tex] ⇒ C -1 = [Tex]\frac{1}{(30\times 1) – (7\times 8)} \begin{bmatrix} 1 &-7\\ -8 & 30 \end{bmatrix}[/Tex] ⇒ C -1 = [Tex]\frac{1}{30 – 56} \begin{bmatrix} 1 &-7\\ -8 & 30 \end{bmatrix}[/Tex] ⇒ C -1 = [Tex]\frac{1}{ – 26} \begin{bmatrix} 1 &-7\\ -8 & 30 \end{bmatrix}[/Tex]

Example 3: Find the inverse of 2×2 matrix D = [Tex]\begin{bmatrix} 1& 4\\ 6 & 12 \end{bmatrix}[/Tex] using elementary operations method.

To find the inverse of D = [Tex]\begin{bmatrix} 1& 4\\ 6 & 12 \end{bmatrix}[/Tex] we will use elementary row operations. D = ID ⇒ [Tex]\begin{bmatrix} 1& 4\\ 6 & 12 \end{bmatrix} [/Tex] = [Tex]\begin{bmatrix} 1& 0\\ 0 & 1 \end{bmatrix} [/Tex] D R 2 → R 2 – 6R 1 ⇒ [Tex]\begin{bmatrix} 1& 4\\ 0 & -12 \end{bmatrix}[/Tex] = [Tex]\begin{bmatrix} 1& 0\\ – 6& 1 \end{bmatrix}[/Tex] D R 2 → R 2 / (-12) ⇒ [Tex]\begin{bmatrix} 1& 4\\ 0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin{bmatrix} 1& 0\\ 1/2 & -1/12 \end{bmatrix}[/Tex] D R 1 → R 1 – 4R 2 ⇒ [Tex]\begin{bmatrix} 1& 0\\ 0 & 1 \end{bmatrix} [/Tex] = [Tex]\begin{bmatrix} -1& 1/3\\ 1/2 & -1/12 \end{bmatrix} [/Tex] D D -1 = [Tex] \begin{bmatrix} -1& 1/3\\ 1/2 & -1/12 \end{bmatrix} [/Tex]

Example 4: Find the inverse of 2×2 matrix E = [Tex]\begin{bmatrix} 1& 5\\ 8 & 2 \end{bmatrix}[/Tex] using elementary operations method.

To find the inverse of E = [Tex] \begin{bmatrix} 1& 5\\ 8 & 2 \end{bmatrix} [/Tex] we will use elementary row operation. E = IE ⇒ [Tex] \begin{bmatrix} 1& 5\\ 8 & 2 \end{bmatrix} [/Tex] = [Tex] \begin{bmatrix} 1& 0\\ 0 & 1 \end{bmatrix} [/Tex] E R 2 → R 2 – 8R 1 ⇒ [Tex] \begin{bmatrix} 1& 5\\ 0 & -38 \end{bmatrix} [/Tex] = [Tex] \begin{bmatrix} 1& 0\\ -8 & 1 \end{bmatrix} [/Tex] E R 2 → R 2 / (-38) ⇒ [Tex] \begin{bmatrix} 1& 5\\ 0 & 1 \end{bmatrix} [/Tex] = [Tex]\begin{bmatrix} 1& 0\\ 8/38 & -1/38 \end{bmatrix}[/Tex] E R 1 → R 1 – 5R 2 ⇒ [Tex] \begin{bmatrix} 1& 0\\ 0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin{bmatrix} -2/38& 5/38\\ 8/38 & -1/38 \end{bmatrix}[/Tex] E So, the inverse of matrix E i.e., E -1 = [Tex] \begin{bmatrix} -2/38& 5/38\\ 8/38 & -1/38 \end{bmatrix} [/Tex]

## Practice Questions on Inverse of 2×2 Matrix

Q1: Find the inverse of 2×2 matrix P = [Tex]\begin{bmatrix} 5 & 22\\ 1 & 9 \end{bmatrix}[/Tex] using inverse formula.

Q2: Find the inverse of 2×2 matrix Q = [Tex]\begin{bmatrix} 4 &2\\ 8 & 13 \end{bmatrix}[/Tex] using inverse formula.

Q3: Find the inverse of 2×2 matrix R = [Tex]\begin{bmatrix} 7& 12\\ 4 & 2 \end{bmatrix}[/Tex] using elementary operations method.

Q4: Find the inverse of 2×2 matrix S = [Tex]\begin{bmatrix} 15 & 90\\ 30 & 6 \end{bmatrix}[/Tex] using elementary operations method.

## FAQs on Inverse of 2×2 Matrix

What is the inverse of two square matrix.

The inverse of two square matrix is obtained by dividing adjoint of matrix by determinant of the matrix.

## What is an Invertible 2×2 Matrix?

The 2×2 matrix whose inverse exists is called an invertible 2×2 matrix.

## How to solve 2×2 Matrix Inverse?

To solve 2×2 matrix to find inverse can be done in two ways. By Inverse Formula By elementary row or column operations

## What is the Formula for Det of 2×2 Matrix?

The formula for the det of 2×2 matrix A = [Tex] \begin{bmatrix} a &b\\ c & d \end{bmatrix}[/Tex] is ad – bc .

## What 2×2 Matrix has no Inverse?

The 2×2 matrix with determinant zero has no inverse matrix.

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Finding and Evaluating Inverse Functions. Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases. Inverting Tabular Functions. Suppose we want to find the inverse of a function represented in table form.

Algebra 2H U2 L5 Inverse Function Extra Practice ©j A2\0e1G8n \K_uTtBaO ESlo_fptwwpafr_eV rLALkCB.g L XAnlHlj Ir]iNgGhCtTsu wr^eOsVeLr_vTeIdu. ... Find the inverse of each function. 1) f (x) = -2x3 + 3 f-1 (x) = 3-x + 3 2 2) g (x) = -x + 4 g-1 (x) = -x + 4 3) g (x) = 3 x + 2 g-1 (x) = (x - 2) 3 4) h (x) = -2x + 5 h-1 (x) = - 1 2 x + 5 2

Graph the inverse of y = 2 x + 3. Consider the straight line, y = 2x + 3, as the original function. It is drawn in blue . If reflected over the identity line, y = x, the original function becomes the red dotted graph. The new red graph is also a straight line and passes the vertical line test for functions. The inverse relation of y = 2 x + 3 ...

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The steps to find inverse of 2×2 matrix by inverse formula is listed. Step 1: First, find the determinant of 2×2 matrix. Step 2: Then, find the adjoint of the 2×2 matrix. Step 3: Put the values of determinant and adjoint of the 2×2 matrix in formula: A-1 = adj (A) / |A|. Step 4: Then, we get the inverse formula of 2×2 matrix.