Inverse Functions

An inverse function undoes the mapping of a one-to-one function. Understanding inverse functions, their domains, and their graphical reflection across $y = x$ is crucial for logarithmic and inverse trigonometric derivatives.

Quick Reference

Property	Formula	Description
Definition	$f^{- 1} (y) = x ⟺ f (x) = y$	Reverses input and output
Domain/Range Swap	$Dom (f^{- 1}) = Rng (f)$	$Rng (f^{- 1}) = Dom (f)$
Cancellation 1	$f^{- 1} (f (x)) = x$	For all $x \in Dom (f)$
Cancellation 2	$f (f^{- 1} (y)) = y$	For all $y \in Dom (f^{- 1})$
Graph Symmetry	Reflection across $y = x$	Point $(a, b) \to (b, a)$

Definition of Inverse Function

Let $f$ be a one-to-one function with domain $A$ and range $B$ . Then its inverse function $f^{- 1}$ has domain $B$ and range $A$ and is defined by:

f^{- 1} (y) = x ⟺ f (x) = y

for any $y$ in $B$ .

Domain and Range Swap:

Dom (f) = Rng (f^{- 1}), Rng (f) = Dom (f^{- 1})

Cancellation Equations:

\begin{aligned} f^{-1}(f(x)) &= x \quad\text{for } x \in \text{Dom}(f) \\ f(f^{-1}(x)) &= x \quad\text{for } x \in \text{Dom}(f^{-1}) \end{aligned}

Caution: $f^{- 1} (x) \neq \frac{1}{f (x)}$ . The reciprocal is written as $[f (x)]^{- 1}$ .

Given that $f$ has an inverse and $f (1) = 3$ , $f (2) = - 4$ , and $f (5) = - 1$ , find $f^{- 1} (3)$ , $f^{- 1} (- 4)$ , and $f^{- 1} (- 1)$ .

Solution

Using $f^{- 1} (y) = x ⟺ f (x) = y$ :

\begin{aligned} f^{-1}(3) &= 1 \quad\text{because } f(1) = 3 \\ f^{-1}(-4) &= 2 \quad\text{because } f(2) = -4 \\ f^{-1}(-1) &= 5 \quad\text{because } f(5) = -1 \end{aligned}

Graphs of Inverse Functions

If $(a, b)$ lies on the graph of $f$ , then $(b, a)$ lies on the graph of $f^{- 1}$ .

The graphs of a function $y = f (x)$ and its inverse $y = f^{- 1} (x)$ are mirror reflections of each other across the line y = x.

Graph reflection across line y = x for f and f inverse — Reflecting $y = f (x)$ across $y = x$ gives the graph of $y = f^{- 1} (x)$ .

For example, $h (x) = \sqrt{2 x + 3}$ ( $x \geq - 3 / 2$ ) and $h^{- 1} (x) = \frac{1}{2} (x^{2} - 3)$ ( $x \geq 0$ ) are reflected across $y = x$ .

Graphs of square root function and quadratic inverse reflected across line y = x — Graphs of $h (x) = \sqrt{2 x + 3}$ and $h^{- 1} (x) = \frac{1}{2} (x^{2} - 3)$ ( $x \geq 0$ ).

Quick Reference

Definition of Inverse Function

Graphs of Inverse Functions

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