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.

Concept

Inverse of a Function

The inverse of a function reverses its $x$ - $y$ coordinates. For a function $f$ , if $x$ is the input and $y$ is the corresponding output, then for the inverse $f^{- 1}, y$ is the input and $x$ would be the corresponding output.

$f (x) = y \Leftrightarrow f^{- 1} (y) = x$

Some function families are inverses of each other. This is because some functions undo each other. For example,

x^{3}

and

3 x

are inverses because radicals and exponents (with the same index) undo each other.

Method

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

f (x) = \frac{2 x - 1}{3} .

1

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. $f (x) = \frac{2 x - 1}{3} \Leftrightarrow y = \frac{2 x - 1}{3}$

2

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. $y = \frac{2 x - 1}{3} \Rightarrow x = \frac{2 y - 1}{3}$

3

Solve for

y

Solve the resulting equation from Step 2 for

y .

Here, this will involve using the inverse operations.

x = \frac{2 y - 1}{3}

MultEqn

$LHS \cdot 3 = RHS \cdot 3$

3 x = 2 y - 1

AddEqn

$LHS + 1 = RHS + 1$

3 x + 1 = 2 y

DivEqn

$LHS / 2 = RHS / 2$

\frac{3 x + 1}{2} = y

RearrangeEqn

Rearrange equation

y = \frac{3 x + 1}{2}

4

Replace

y

with

f^{- 1} (x)

Just as $f (x) = y$ shows the input-output relationship of $f,$ so does $f^{- 1} (x) = y .$ Thus, replacing $y$ with $f^{- 1} (x)$ gives the rule for the inverse of $f .$

$y = \frac{3 x + 1}{2} \Leftrightarrow f^{- 1} = \frac{3 x + 1}{2}$

Notice that in $f,$ the input is multiplied by $2,$ decreased by $1$ and divided by $3 .$ From the rule of $f^{- 1},$ 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 $2 .$

Determine the inverse of the function

fullscreen

Exercise

Some of the coordinates of the function $g$ are shown in the table. Find $g^{- 1},$ then graph $g$ and $g^{- 1}$ on the same coordinate plane.

$x$	$g (x)$
$- 4$	$3$
$- 2$	$2$
$0$	$1$
$2$	$0$
$4$	$- 1$

Show Solution

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 $g^{- 1} .$ The following table describes $g^{- 1} (x) .$

$x$	$g^{- 1} (x)$
$3$	$- 4$
$2$	$- 2$
$1$	$0$
$0$	$2$
$- 1$	$4$

We can graph both $g$ and $g^{- 1}$ by marking the points from both tables on the same coordinate plane.

Concept

Graphs of Functions and Inverses

The graphs of a function and its inverse have a noteworthy relationship. In the coordinate plane, the function $f (x) = 2 x - 3 and its inverse f^{- 1} (x) = \frac{x + 3}{2}$ are graphed. Notice that the points of $f^{- 1}$ are the reversed points of $f .$

Because the coordinates of the points are reversed, $f^{- 1} (x)$ is a reflection of $f (x)$ in the line $y = x .$

This is true for all functions and their inverses.

Concept

Invertible Function

A function is said to be invertible if its inverse is also a function. An example of an invertible function is $f (x) = 0.5 x - 2,$ because its inverse, $f^{- 1} (x) = 2 x + 4,$ is a linear function. Consider the function $g (x) = x^{2},$ whose inverse is $\pm x .$

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 $x \geq 0,$ the inverse, $g^{- 1} (x) = x,$ is a function.

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

Method

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, $each y must cor exactly o - value respond to ne x - value .$ 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

1

2

3

4

Example

Determine the inverse of the function