Relating a Function with Its Inverse
If the result of switching the inputs and outputs of a function is also a function, it is called the inverse function. The composition of a function and its inverse is the identity function.The inverse of the function is written as . This is read " inverse of ." This notation does not mean the same thing as:
Graphs of Inverse Functions
Restricting the Domain
The inverse of a function is defined only if the function is one-to-one. One way to test whether a function is one-to-one is the horizontal line test. If the function does not pass the horizontal line test, then the inverse of the function is not a function.
Analyzing the graph of a function can help determine restrictions on the domain that will make the function one-to-one.
Horizontal and Vertical Line Test
|The function is a function, but it does not pass the horizontal line test, so it is not one-to-one. This means the inverse of the function is not a function.|
|The graph of the inverse of is a reflection of over the line .
To identify the equation of the inverse, replace with , and then switch the positions of and .
The graph of the inverse does not pass the vertical line test, so it is not a function.
|If the domain of the function is restricted to , the function passes the horizontal line test, so it is one-to-one.|
|When the domain of is restricted to , the graph of the inverse passes the vertical line test, so it is a function.
To identify the rule for the function, solve the inverse for .
Since the domain of the original function was restricted to positive values of , the inverse function is only the positive square root, . Replace with .
Restrict the domain so that the new function is one-to-one.Limit the -values used so that each -value appears only once.
So is one-to-one for .
Replace with , and include the domain.For , the inverse is: