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Miscellaneous Functions

Describing Inverses of Functions

The inverse of a function can itself be a function. In this section, properties of inverses and invertible functions will be explored.


Inverse of a Function

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.


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


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.


Graphs of Functions and Inverses

The graphs of a function and its inverse have a noteworthy relationship. In the coordinate plane, the function are graphed. Notice that the points of are the reversed points of

Because the coordinates of the points are reversed, is a reflection of in the line

This is true for all functions and their inverses.


Invertible Function

A function is said to be invertible if its inverse is also a function. An example of an invertible function is because its inverse, is a linear function. Consider the function whose inverse is

Notice that the inverse fails the Vertical Line Test. Thus, it is not a function, which, in turn means that is not invertible. However, if the domain of is restricted to the inverse, is a function.

Thus, some functions that are not inherently invertible can be made invertible by restricting their domains.


Horizontal Line Test

The Vertical Line Test determines if the graph of a relation is a function, or if each -value corresponds with exactly one -value. For a function to be invertible, This leads to the Horizontal Line Test, which is performed by moving an imaginary horizontal line across the graph of a function. If this line intersects the graph more than once anywhere, the function is not invertible.

Perform test

Change function

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