Math Core

If a function turns Celsius into Fahrenheit, you'll eventually want a function that turns Fahrenheit back into Celsius. A function that undoes another one is its inverse. Inverses tie this unit together (roots undo powers) and set up the next one, where logarithms are defined as the inverses of exponential functions.

Undoing a function

Consider $f(x) = 2x + 3$ . It takes an input, multiplies by 2, then adds 3. To undo it, undo the steps in reverse order: subtract 3, then divide by 2. That gives

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

Try it: $f(5) = 13$ , and $g(13) = \dfrac{13 - 3}{2} = 5$ . The function $g$ took you right back where you started. Composition makes this precise.

Definition

Inverse functions

Functions $f$ and $g$ are inverses of each other if

f(g(x)) = x \quad \text{for every } x \text{ in the domain of } g \qquad \text{and} \qquad g(f(x)) = x \quad \text{for every } x \text{ in the domain of } f.

The inverse of $f$ is written $f^{-1}$ , read " $f$ inverse."

For the example above: $f(g(x)) = 2\left(\dfrac{x - 3}{2}\right) + 3 = x - 3 + 3 = x$ and $g(f(x)) = \dfrac{(2x + 3) - 3}{2} = x$ . Both compositions give $x$ , so $g = f^{-1}$ .

Common mistake

The $-1$ in $f^{-1}$ is not an exponent. $f^{-1}(x)$ means the inverse function, not $\dfrac{1}{f(x)}$ . For $f(x) = 2x + 3$ , $f^{-1}(x) = \dfrac{x - 3}{2}$ , while $\dfrac{1}{f(x)} = \dfrac{1}{2x + 3}$ , a completely different function.

Inverses swap inputs and outputs

If $f(a) = b$ , then $f^{-1}(b) = a$ . So every point $(a, b)$ on the graph of $f$ becomes the point $(b, a)$ on the graph of $f^{-1}$ . Two consequences follow:

The domain of $f^{-1}$ is the range of $f$ , and the range of $f^{-1}$ is the domain of $f$ .
Swapping coordinates reflects a point across the line $y = x$ . So the graph of $f^{-1}$ is the reflection of the graph of $f$ across $y = x$ .

f(x) = x³ + 1 and its inverse f⁻¹(x) = ∛(x − 1) are mirror images across the dashed line y = x. The point (1, 2) on f matches (2, 1) on f⁻¹.Open in grapher →

Finding an inverse algebraically

Since an inverse swaps the roles of $x$ and $y$ , you can find it by literally swapping them.

Finding the inverse of a function

Replace $f(x)$ with $y$ .
Swap $x$ and $y$ .
Solve the new equation for $y$ .
Replace $y$ with $f^{-1}(x)$ . State any domain restriction, and check by composition if you like.

Worked example: Inverses of linear and cubic functions

Find the inverse of each function.

$f(x) = \dfrac{x}{4} - 5$
$f(x) = 2x^3 + 1$

Solutions.

Write $y = \dfrac{x}{4} - 5$ and swap: $x = \dfrac{y}{4} - 5$ . Solve: $x + 5 = \dfrac{y}{4}$ , so $y = 4(x + 5) = 4x + 20$ . Thus $f^{-1}(x) = 4x + 20$ .
Write $y = 2x^3 + 1$ and swap: $x = 2y^3 + 1$ . Solve for $y$ :

\begin{aligned} x - 1 &= 2y^3 \\ \frac{x - 1}{2} &= y^3 \\ y &= \sqrt[3]{\frac{x - 1}{2}} \end{aligned}

So $f^{-1}(x) = \sqrt[3]{\dfrac{x - 1}{2}}$ . Check one value: $f(1) = 3$ , and $f^{-1}(3) = \sqrt[3]{1} = 1$ . ✓

When does a function have an inverse?

Not every function can be undone. For $f(x) = x^2$ , both $f(3)$ and $f(-3)$ equal $9$ . An "inverse" would have to send $9$ back to both $3$ and $-3$ , and a function can't give two outputs for one input.

A function has an inverse function exactly when it is one-to-one: different inputs always give different outputs. On a graph, that means no horizontal line crosses the graph more than once.

The horizontal line test

A function has an inverse that is a function if and only if no horizontal line intersects its graph more than once. Such a function is called one-to-one.

Lines with nonzero slope, $y = x^3$ , and $y = \sqrt{x}$ pass the test. Parabolas, $y = |x|$ , and $y = x^4$ fail it.

Restricting the domain

You can often make a function one-to-one by keeping only part of its domain. If you restrict $f(x) = x^2$ to $x \ge 0$ , the graph is half a parabola, which passes the horizontal line test, and its inverse is $f^{-1}(x) = \sqrt{x}$ . This is exactly why $\sqrt{\ \ }$ gives only the nonnegative root: it undoes squaring on the restricted domain $x \ge 0$ .

Worked example: An inverse on a restricted domain

Find the inverse of $f(x) = (x - 3)^2$ for $x \ge 3$ .

The restriction makes $f$ one-to-one. Its domain is $x \ge 3$ and its range is $y \ge 0$ .

Swap: $x = (y - 3)^2$ . Take square roots. Because $y \ge 3$ (the old domain becomes the new range), $y - 3 \ge 0$ , so use the positive root only:

\sqrt{x} = y - 3, \qquad y = \sqrt{x} + 3.

So $f^{-1}(x) = \sqrt{x} + 3$ , with domain $x \ge 0$ (the range of $f$ ).

f(x) = (x − 3)² for x ≥ 3 and its inverse f⁻¹(x) = √x + 3, reflected across the dashed line y = x.Open in grapher →

Inverses of radical functions

A radical function's inverse is a power function, but the domain restriction carries over. Take $f(x) = \sqrt{x + 4}$ . Its domain is $x \ge -4$ and its range is $y \ge 0$ . Swapping and solving: $x = \sqrt{y + 4}$ , so $x^2 = y + 4$ and $y = x^2 - 4$ . Since the range of $f$ is $y \ge 0$ , the inverse is

f^{-1}(x) = x^2 - 4, \quad x \ge 0.

Without the restriction $x \ge 0$ , you'd have a full parabola, which is not the inverse of $f$ .

Tip

After finding an inverse, pick a simple input, run it through $f$ , then run the output through your $f^{-1}$ . If you get the original input back, your inverse is almost certainly right.

Practice

Practice 1

Find $f^{-1}(x)$ for $f(x) = 4x + 7$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 2

A one-to-one function $f$ has $f(1) = 4$ , $f(3) = -2$ and $f(5) = 0$ . Find $f^{-1}(0)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Find $f^{-1}(x)$ for $f(x) = \dfrac{x}{3} - 2$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 4

Find $f^{-1}(x)$ for $f(x) = \sqrt[3]{x + 5}$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 5

Find $f^{-1}(x)$ for $f(x) = 2x^3 - 3$ . (Type a cube root as cbrt( ).)

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 6

Which function has an inverse that is also a function?

Practice 7

Let $f(x) = \sqrt{x - 2}$ with domain $x \ge 2$ . Which is $f^{-1}$ ?

Practice 8

Find $f^{-1}(x)$ for $f(x) = \dfrac{2x + 1}{x - 3}$ .

Enter an expression, e.g. 3x^2 - 2x + 1