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 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 $f^{- 1}$ such that they undo each other.

$f (f^{- 1} (x)) = x and f^{- 1} (f (x)) = x$

Also, if x is the input of a function f and y its corresponding output, then y is the input of $f^{- 1}$ and x its corresponding output.

Example

Consider a function f and its inverse

f^{- 1} .

f (x) = 2 x - 3 and f^{- 1} (x) = \frac{x + 3}{2}

These functions will be shown to undo each other. To do so, it needs to be proven that

f (f^{- 1} (x)) = x

and that

f^{- 1} (f (x)) = x .

To start, the first equality will be proven. First, the definition of f will be used.

Now, in the above equation,

\frac{x + 3}{2}

will be substituted for

f^{- 1} (x) .

2 f^{- 1} (x) - 3 = ? x

Substitute

$f^{- 1} (x) = \frac{x + 3}{2}$

2 (\frac{x + 3}{2}) - 3 = ? x

Simplify left-hand side

DenomMultFracToNumber

$2 \cdot \frac{a}{2} = a$

(x + 3) - 3 = ? x

RemovePar

Remove parentheses

x + 3 - 3 = ? x

SubTerm

Subtract term

x = x ✓

A similar procedure can be performed to show that

f^{- 1} (f (x)) = x .

	Definition of First Function	Substitute Second Function	Simplify
$f (f^{- 1} (x)) = ? x$	$2 f^{- 1} (x) - 3 = ? x$	$2 (\frac{x + 3}{2}) - 3 = ? x$	$x = x ✓$
$f^{- 1} (f (x)) = ? x$	$\frac{f ( x ) + 3}{2} = ? x$	$\frac{2 x - 3 + 3}{2} = ? x$	$x = x ✓$

Therefore, f and $f^{- 1}$ 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 $f^{- 1}$ are the reversed points on the graph of f.

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

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.

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.

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⋅3=RHS⋅3

3x=2y−1

AddEqn

LHS+1=RHS+1

3x+1=2y

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.

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)=0.5x−2,

because its inverse,

f^{- 1} (x) = 2 x + 4,

is a linear function. Consider the function g(x)=x2, 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≥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