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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 Function

Every function has an inverse relation. If this inverse relation is also a function, then it is called an inverse function. In other words, the inverse of a function f is another function such that they undo each other.

Also, if x is the input of a function f and y its corresponding output, then y is the input of and x its corresponding output.


Consider a function f and its inverse
These functions will be shown to undo each other. To do so, it needs to be proven that and that To start, the first equality will be proven. First, the definition of f will be used.
Now, in the above equation, will be substituted for
Simplify left-hand side
A similar procedure can be performed to show that
Definition of First Function Substitute Second Function Simplify

Therefore, f and undo each other. The graphs of these functions are each other's reflection across the line y=x. This means that the points on the graph of are the reversed points on the graph of f.

inverse function


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 f(x) with y
To begin, since f(x)=y describes the input-output relationship of the function, replace f(x) with y in the function rule.


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


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


Replace y with

Just as f(x)=y shows the input-output relationship of f, so does Thus, replacing y with gives the rule for the inverse of f.

Notice that in f, the input is multiplied by 2, decreased by 1 and divided by 3. From the rule of it can be seen that x undergoes the inverse of these operation in the reverse order. Specifically, x is multiplied by 3, increased by 1, and divided by 2.


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

x g(x)
-4 3
-2 2
0 1
2 0
4 -1
Show Solution

An inverse of a function reverses its x- and y-coordinates. When a function is expressed as a table of values, finding its inverse means switching the coordinates. For example, the point (-4,3) on g becomes (3,-4) on The following table describes

3 -4
2 -2
1 0
0 2
-1 4

We can graph both g 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 f.

Because the coordinates of the points are reversed, is a reflection of f(x) in the line y=x.

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 g(x)=x2, whose inverse is

Notice that the inverse fails the Vertical Line Test. Thus, it is not a function, which, in turn means that g is not invertible. However, if the domain of g is restricted to x0, 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 x-value corresponds with exactly one y-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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