Inverse Functions

1 min read

This concept page covers the theory and applications of inverse functions.

Definition

A function $f$ has an inverse function $f^{-1}$ if and only if $f$ is one-to-one (injective).

If $f^{-1}$ exists, then:

$f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$
$f(f^{-1}(y)) = y$ for all $y$ in the domain of $f^{-1}$

One-to-One Functions

A function $f$ is one-to-one if different inputs produce different outputs:

$$f(x_1) = f(x_2) \implies x_1 = x_2$$

Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once.

Finding Inverse Functions

To find $f^{-1}$:

Write $y = f(x)$
Solve for $x$ in terms of $y$
Swap $x$ and $y$ to get $y = f^{-1}(x)$

Example: Find the inverse of $f(x) = 2x + 3$.

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

Properties

Domain and Range

Domain of $f^{-1}$ = Range of $f$
Range of $f^{-1}$ = Domain of $f$

Graphical Relationship

The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$.

Derivative of Inverse Functions

If $f$ is differentiable and $f'(x) \neq 0$:

$$\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}$$

Important Inverse Functions

Function	Inverse	Domain Restriction
$e^x$	$\ln x$	$x > 0$
$\sin x$	$\arcsin x$	$[-1, 1]$
$\cos x$	$\arccos x$	$[-1, 1]$
$\tan x$	$\arctan x$	$\mathbb{R}$

Common Misconceptions

Common misconception

$f^{-1}(x)$ means $\frac{1}{f(x)}$, the reciprocal of $f$.

This is the concept-image-conflicts-definition error. The superscript $-1$ in $f^{-1}$ denotes the inverse function, not a negative exponent. The reciprocal of $f(x)$ is written $[f(x)]^{-1}$ or $\frac{1}{f(x)}$, and it is a completely different object. For example, if $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x-3}{2}$, while $\frac{1}{f(x)} = \frac{1}{2x+3}$. These are not equal for any $x$. The inverse function reverses the input-output relationship; the reciprocal simply takes the multiplicative inverse of the output value.

← Skills Index