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 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 after substituting a value for 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 we represent its inverse as read as inverse of The raised is part of the notation. It is not an exponent; it does not imply a power of . In other words, does not mean because is the reciprocal of and not the inverse.
The “exponent-like” notation comes from an analogy between function composition and multiplication: just as (1 is the identity element for multiplication) for any nonzero number so equals the identity function, that is,
This holds for all in the domain of 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 we can verify whether some other function is the inverse of by checking whether either or is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)
For example, and are inverse functions.
A few coordinate pairs from the graph of the function are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function 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.
For any one-to-one function a function is an inverse function of if This can also be written as for all in the domain of It also follows that for all in the domain of if is the inverse of
The notation is read inverse.” Like any other function, we can use any variable name as the input for so we will often write which we read as inverse of 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 and 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 then and
Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed. See Table 1.
Given that what are the corresponding input and output values of the original function
Given two functions and test whether the functions are inverses of each other.
- Determine whether or
- If both statements are true, then and If either statement is false, then both are false, and and
Testing Inverse Relationships Algebraically
If and is
This is enough to answer yes to the question, but we can also verify the other formula.
Notice the inverse operations are in reverse order of the operations from the original function.
If and is
Determining Inverse Relationships for Power Functions
If (the cube function) and is
No, the functions are not inverses.
The correct inverse to the cube is, of course, the cube root that is, the one-third is an exponent, not a multiplier.
Finding Domain and Range of Inverse Functions
The outputs of the function are the inputs to so the range of is also the domain of Likewise, because the inputs to are the outputs of the domain of is the range of 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 is because a square “undoes” a square root; but the square is only the inverse of the square root on the domain since that is the range of
We can look at this problem from the other side, starting with the square (toolkit quadratic) function 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 with its domain limited to which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).
If on then the inverse function is
- The domain of = range of =
- The domain of = range of =
Is it possible for a function to have more than one inverse?
No. If two supposedly different functions, say, and both meet the definition of being inverses of another function then you can prove that 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.
The range of a function is the domain of the inverse function
The domain of is the range of
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.
|Reciprocal squared||Cube root||Square root||Absolute value|
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 meaningful inverse.
The absolute value function can be restricted to the domain where it is equal to the identity function.
The reciprocal-squared function can be restricted to the domain
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 is and the range of function is 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 is given in Table 3, showing distance in miles that a car has traveled in minutes. Find and interpret
The inverse function takes an output of and returns an input for So in the expression 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function 90 minutes, so 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 then By this definition, if we are given then we are looking for a value so that In this case, we are looking for a so that which is when
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 is given in Figure 5. Find and
To evaluate we find 3 on the x-axis and find the corresponding output value on the y-axis. The point tells us that
To evaluate recall that by definition means the value of x for which By looking for the output value 3 on the vertical axis, we find the point on the graph, which means so by definition, See Figure 6.
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, as a function of we can often find the inverse function by solving to obtain as a function of
Given a function represented by a formula, find the inverse.
- Make sure is a one-to-one function.
- Solve for
- Interchange and
- Replace with . (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 to represent the conversion because the input and output variables are descriptive, and writing could get confusing.
Solve for in terms of given
Solving to Find an Inverse Function
Find the inverse of the function
The domain and range of exclude the values 3 and 4, respectively. and 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
The domain of is Notice that the range of is so this means that the domain of the inverse function is also
The formula we found for looks like it would be valid for all real However, itself must have an inverse (namely, ) so we have to restrict the domain of to in order to make a one-to-one function. This domain of is exactly the range of
What is the inverse of the function 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 restricted to the domain on which this function is one-to-one, and graph it as in Figure 7.
Restricting the domain to 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, What happens if we graph both and on the same set of axes, using the axis for the input to both
We notice a distinct relationship: The graph of is the graph of reflected about the diagonal line 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 in Figure 9, sketch a graph of
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 and range of so the inverse will have a domain of and range of
If we reflect this graph over the line the point reflects to and the point reflects to Sketching the inverse on the same axes as the original graph gives Figure 10.
Is there any function that is equal to its own inverse?
Yes. If then and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because
Any function where is a constant, is also equal to its own inverse.
1.7 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 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 is its own inverse for all real numbers
For the following exercises, find for each function.
For the following exercises, find a domain on which each function is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of restricted to that domain.
- ⓐ Find and
- ⓑ What does the answer tell us about the relationship between and
For the following exercises, use function composition to verify that and are inverse functions.
For the following exercises, use a graphing utility to determine whether each function is one-to-one.
For the following exercises, determine whether the graph represents a one-to-one function.
For the following exercises, use the graph of shown in Figure 11.
For the following exercises, use the graph of the one-to-one function shown in Figure 12.
Sketch the graph of
If the complete graph of is shown, find the domain of
If the complete graph of is shown, find the range of
For the following exercises, evaluate or solve, assuming that the function is one-to-one.
For the following exercises, use the values listed in Table 6 to evaluate or solve.
For the following exercises, find the inverse function. Then, graph the function and its inverse.
Find the inverse function of Use a graphing utility to find its domain and range. Write the domain and range in interval notation.
To convert from degrees Celsius to degrees Fahrenheit, we use the formula Find the inverse function, if it exists, and explain its meaning.
The circumference of a circle is a function of its radius given by Express the radius of a circle as a function of its circumference. Call this function Find 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, in hours given by Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function Find and interpret its meaning.