Function Properties: Inverses

Definition. [Function's Inverse] The inverse of a function \( f \), denoted \( f^{-1} \), reverses the mapping of \( f \).

This is the exact same concept as for converse relations, in which we took every point $(x,y)$ in the relation and flip it into $(y, x)$ to get the converse.

Theorem. [Inverse] The function $f$ has an inverse if and only if $f$ is a bijection. That is, if a function is not bijective, its inverse does not exist as a function.

If \( f: A \to B \) is a bijection, then the inverse \( f^{-1}: B \to A \) satisfies \( f^{-1}(f(x)) = x \) for all \( x \in A \). Also, \( f(f^{-1}(y)) = y \) for all \( y \in B \).

For example, if \( f(x) = 3x + 2 \), then its inverse is \( f^{-1}(y) = \frac{y - 2}{3} \). Note that $f^{-1}(f(x)) = \frac{(3x + 2) - 2}{3} = \frac{3x}{3} = x$.

An inverse function "undoes" the action of the original function.