This concept page covers the theory and applications of inverse functions.
A function $f$ has an inverse function $f^{-1}$ if and only if $f$ is one-to-one (injective).
If $f^{-1}$ exists, then:
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.
To find $f^{-1}$:
Example: Find the inverse of $f(x) = 2x + 3$.
The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$.
If $f$ is differentiable and $f'(x) \neq 0$:
$$\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}$$
| Function | Inverse | Domain Restriction |
|---|---|---|
| $e^x$ | $\ln x$ | $x > 0$ |
| $\sin x$ | $\arcsin x$ | $[-1, 1]$ |
| $\cos x$ | $\arccos x$ | $[-1, 1]$ |
| $\tan x$ | $\arctan x$ | $\mathbb{R}$ |