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Pre-Calculus 12 · Mathematics · Lesson 1.3

Inverses

Level
Grade 12
Reading time
4 min reading
Review
Last reviewed 2026-09-03
Errata

Abstract

Undo a function by swapping input and output — where inverses come from.

What is an inverse function?

Short answer

An inverse function reverses another function: if f maps x to y, its inverse f⁻¹ maps y back to x. You find it by swapping x and y and solving for y. A function has an inverse only if it is one-to-one (passes the horizontal line test), and the graph of f⁻¹ is the reflection of f across the line y = x.

Concept preview

What you will be able to do

  1. 01Find the inverse of a function algebraically and graphically
  2. 02Use the horizontal-line test and restrict domains to guarantee an inverse function
  3. 03Verify inverses with composition and relate their domains and ranges

The inverse of a function ff reverses what ff does: if ff sends aa to bb, then f−1f^{-1} sends bb back to aa. Formally, f−1(f(x))=xf^{-1}(f(x)) = x and f(f−1(x))=xf(f^{-1}(x)) = x. Because inputs and outputs swap, every point (a,b)(a, b) on y=f(x)y = f(x) becomes (b,a)(b, a) on y=f−1(x)y = f^{-1}(x). Note f−1f^{-1} means inverse, not a reciprocal — it is not 1f\dfrac{1}{f}.

School Plan

Inverses

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Related topics

Logarithms →

Frequently asked questions

When does a function have an inverse?
Only when it is one-to-one — each output comes from exactly one input, which it passes the horizontal line test.
How are a function and its inverse related graphically?
Their graphs are mirror images across the line y = x.