---
title: "Inverse functions"
description: "Find the rule that undoes a function by swapping input and output and rearranging, and see the inverse graph as a reflection in the line y = x."
canonical: https://lightmysky.com/learn/mathematics/inverse-functions-mt_7k7q5ZjMb4
source: https://lightmysky.com/learn/mathematics/inverse-functions-mt_7k7q5ZjMb4.md
retrieved: 2026-11-12
---

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# Inverse functions

Find the rule that undoes a function by swapping input and output and rearranging, and see the inverse graph as a reflection in the line y = x.

Subject: Mathematics · Area: Algebra · Ages 15 to 16
Page: https://lightmysky.com/learn/mathematics/inverse-functions-mt_7k7q5ZjMb4

## Ready when they can

- Find f⁻¹(x) for a linear rule by rearranging
- Check an inverse by showing that ff⁻¹(x) gives back x
- Describe the inverse graph as a reflection of the original in y = x

## Lesson: Inverse functions

You have composed functions by feeding one output into another rule. An inverse undoes a function: applying f and then f⁻¹ returns the original input. Read f⁻¹ as 'f inverse'. The -1 in this notation names the inverse; it does not mean take a reciprocal. An inverse function exists only when each output of f comes from one input.

*(drawing: Two machines that cancel each other out, no matter what you feed in.)*

For a linear rule with nonzero slope, find the inverse by swapping input and output, then rearranging. Write y = 2x + 1. Swap x and y: x = 2y + 1. Subtract 1 to get 2y = x - 1, then divide by 2: y = (x - 1) / 2. Thus f⁻¹(x) = (x - 1) / 2. The old output becomes the new input.

Check a candidate inverse by composing both ways. You need f(f⁻¹(x)) = x for outputs of f and f⁻¹(f(x)) = x for inputs of f. A mismatch reveals an error. The inverse is different from the reciprocal: for f(x) = 2x + 1, f⁻¹(x) = (x - 1) / 2, whereas 1/f(x) = 1/(2x + 1). A check at one number tests that number, not every input.

**Example.** For f(x) = 2x + 1, swap y = 2x + 1 to x = 2y + 1. Rearranging gives f⁻¹(x) = (x - 1) / 2. Check both ways: f(f⁻¹(x)) = 2((x - 1)/2) + 1 = x, and f⁻¹(f(x)) = (2x + 1 - 1)/2 = x. At input 4, f gives 9 and the inverse gives (9 - 1)/2 = 4.

**Example.** For f(x) = (x - 4)/3, write y = (x - 4)/3 and swap: x = (y - 4)/3. Multiply by 3, then add 4: 3x = y - 4, so f⁻¹(x) = 3x + 4. Check f(f⁻¹(x)) = (3x + 4 - 4)/3 = x. The original subtracts 4 and divides by 3. The inverse undoes these operations in reverse order: multiply by 3, then add 4.

**Example.** Axes cross at the origin (0, 0): x horizontal, y vertical. Plot (a, b) from there: a right, then b up; negatives mean left or down. Use equal unit lengths on both axes. The line y = x joins equal-coordinate points. Folding along it swaps (a, b) to (b, a). On f, a is input and b output; swapping gives the inverse graph. For f(x) = 2x + 1, (0, 1) becomes (1, 0), and (2, 5) becomes (5, 2). Nonconstant linear functions have linear inverses. Curves reflect this way when an inverse function exists.

The inverse of f⁻¹ is f. To recover f from its inverse, apply the same swap-and-rearrange method to the supplied inverse rule. Applying one inverse twice is different: evaluate the inverse once, then use that result as its next input. For h(x) = (x + 6)/7, h(8) = 2 and h(h(8)) = h(2) = 8/7.

**Recap.** Find an inverse by swapping x and y and solving for y. Check composition both ways on the relevant inputs. The inverse graph reflects in y = x, swapping each point's coordinates.

## Practice

35 questions on this page, each with its working shown.

## Needs first

- [Composite functions](https://lightmysky.com/learn/mathematics/composite-functions-mt_jqGJHXgYNw)

## Opens up

- [Transformations of Graphs](https://lightmysky.com/learn/mathematics/transformations-of-graphs-mt_691_c-7Z5M)
- [Domain and Range of a Function](https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc)
- [Implicit Differentiation and Derivatives of Inverse Functions](https://lightmysky.com/learn/mathematics/implicit-differentiation-and-derivatives-of-inverse-functions-mt_P4ABBGsKFT)
- [Logarithms as the Inverse of Exponentials](https://lightmysky.com/learn/mathematics/logarithms-as-the-inverse-of-exponentials-mt_skLQLUan0T)