89 Find or evaluate the inverse of a function
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.
Example 5: Interpreting the Inverse of a Tabular Function
A function
30 | 50 | 70 | 90 | |
20 | 40 | 60 | 70 |
Solution
The inverse function takes an output of
Alternatively, recall that the definition of the inverse was that if
Try It 5
Using the table below, find and interpret (a)
30 | 50 | 60 | 70 | 90 | |
20 | 40 | 50 | 60 | 70 |
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.
How To: 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.
Example 6: Evaluating a Function and Its Inverse from a Graph at Specific Points
A function

Solution
To evaluate
To evaluate

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,
How To: Given a function represented by a formula, find the inverse.
- Make sure
is a one-to-one function. - Solve for
. - Interchange
and .
Example 7: Inverting the Fahrenheit-to-Celsius Function
Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.
Solution
By solving in general, we have uncovered the inverse function. If
then
In this case, we introduced a function
Example 8: Solving to Find an Inverse Function
Find the inverse of the function
Solution
So
Example 9: Solving to Find an Inverse with Radicals
Find the inverse of the function
Solution
So
The domain of
Analysis of the Solution
The formula we found for
Try It 8
What is the inverse of the function
Analysis of the Solution
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 the table below.