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Miscellaneous Functions and Equations

Finding Inverses of Functions

The inverse of a function reverses its - coordinates. For a function , if is the input and is the corresponding output, then for the inverse is the input and would be the corresponding output.

Some function families are inverses of each other. This is because some functions undo each other. For example, and are inverses because radicals and exponents (with the same index) undo each other.

Some of the coordinates of the function are shown in the table. Find then graph and on the same coordinate plane.

Show Solution

An inverse of a function reverses its - and -coordinates. When a function is expressed as a table of values, finding its inverse means switching the coordinates. For example, the point on becomes on The following table describes

We can graph both and by marking the points from both tables on the same coordinate plane.


Finding the Inverse of a Function

Depending on how a function is presented, finding its inverse can be done in different ways. When a function's rule is given, finding the inverse algebraically is advantageous. Consider the function


Replace with

To begin, since describes the input-output relationship of the function, replace with in the function rule.


Switch and

Because the inverse of a function reverses and the variables can be switched. Notice that every other piece in the function rule remains the same.


Solve for
Solve the resulting equation from Step 2 for Here, this will involve using the inverse operations.


Replace with

Just as shows the input-output relationship of so does Thus, replacing with gives the rule for the inverse of

Notice that in the input is multiplied by decreased by and divided by From the rule of it can be seen that undergoes the inverse of these operation in the reverse order. Specifically, is multiplied by increased by and divided by


Consider the quadratic function Find its inverse function for when

Show Solution
To find the inverse of we first need to replace with The next step is to switch and in the function rule. Now we need to solve for The resulting equation will be the inverse of the given function.
Switching and also switches the domain and the range. This means that the restriction on the domain of the function, becomes a restriction on the range of the inverse function, Thus, there are no negative -values. Finally, to indicate that this is the inverse of we will replace with
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