Inverse of a Function

6. Inverse of a Function

The Key Theorem

Why bijectivity is exactly what's needed: to define $f^{-1}(y)$ unambiguously for every $y \in B$, two things must hold. First, $f$ must be onto — otherwise some $y \in B$ has no $x$ with $f(x) = y$, leaving $f^{-1}(y)$ undefined. Second, $f$ must be one-one — otherwise some $y$ would have two different preimages $x_1 \neq x_2$ with $f(x_1) = f(x_2) = y$, and $f^{-1}(y)$ could not be assigned a single value. Bijectivity guarantees each $y \in B$ has exactly one preimage, which is precisely what $f^{-1}(y)$ names.

Method — Finding the Inverse of a Bijective Function

Write $y = f(x)$, solve algebraically for $x$ in terms of $y$, then relabel the variable: this expression is $f^{-1}(x)$.

Illustration. Let $f : \mathbb{R} \to \mathbb{R}$, $f(x) = \dfrac{3x+2}{5}$ (a bijection, being a linear function with non-zero slope).

Verification:

$f(f^{-1}(x)) = \frac{3\left(\frac{5x-2}{3}\right)+2}{5} = \frac{(5x-2)+2}{5} = \frac{5x}{5} = x \; \checkmark$ …