Recipe

Find the inverse of a function

Use when: you're given a function f and asked to find its inverse f^{-1}

Goal: the inverse function's formula, or a clear reason no inverse function exists

Taught in Inverse functions

1. 1

   Check that f is both one-to-one and onto its stated codomain before doing any algebra.

   Example

   f: \mathbb{R} \to \mathbb{R}, f(x) = x^2, is not one-to-one, since f(-3) = f(3) = 9. Two different inputs share an output.

   • f is one-to-one and onto — continue to the next step.
   • f fails one-to-one or onto — stop here; no inverse function exists at all.

   Why: an inverse function must send each output back to exactly one input, which only a bijection guarantees.

2. 2

   Replace f(x) with y, then rearrange the equation so x sits alone on one side.

   Example

   For f(x) = 2x+5: write y = 2x+5, subtract 5 to get y-5=2x, then divide by 2 to get x = \dfrac{y-5}{2}.

   Why: reversing which variable is isolated is exactly what "undoing" f means algebraically.

3. 3

   The formula just solved for x is the rule that recovers the input from a given output; write it as f^{-1}(y).

   Example

   f^{-1}(y) = \dfrac{y-5}{2}.

   Why: the definition of f^{-1} is exactly "the rule sending each output back to the input that produced it," which is what solving for x already computed.

4. 4

   Check that f^{-1}(f(x)) = x using the derived formula, catching any algebra slip before the result gets used elsewhere.

   Example

   f^{-1}(f(x)) = f^{-1}(2x+5) = \dfrac{(2x+5)-5}{2} = \dfrac{2x}{2} = x. The rewind lands back exactly on x.

   Why: composing catches an algebra slip that a quick glance at the formula alone would miss.