--- 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: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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 Last stop you chained two rules: the output of one fed the next, and you saw that fg(x) and gf(x) usually differ. An inverse is a very special second rule. It is the rule that undoes the first. Feed any input through f and then through f to the power of -1, of x, and you are back where you started. That minus one is a label, not a power: say the symbol as 'f inverse'. The distinction carries its weight in this topic. *(drawing: Two machines that cancel each other out, no matter what you feed in.)* How do you find the inverse of a linear rule? Swap and rearrange. Start from y = 2x + 1. Swap the x and the y to get x = 2y + 1. Now rearrange to make y the subject, exactly as you did with formulae: 2y = x - 1, so y = (x - 1) / 2. That new rule is the inverse. The swap is the whole idea in one move. In the original, x is the input and y is the output. After the swap, the old output is the input, which is exactly what an undoing should do. The check is a composition. Feed f into its inverse and you should get x back, and the other way round too. If f of f inverse of x, gives x, the rearranging was right. If it gives something else, hunt for the slip. The classic trap: f inverse of x, is not 1 over f of x. The minus one is a label, not an exponent, so the inverse is not the reciprocal. The reciprocal of 2x + 1 is a fraction, 1 over (2x + 1). The inverse is (x - 1) / 2. Different animals, one notation. **Example.** Find the inverse of f(x) = 2x + 1, and check it. Write y = 2x + 1. Swap: x = 2y + 1. Rearrange: 2y = x - 1, so y = (x - 1) / 2, the inverse rule. Check both ways: f of the inverse of x is 2 times (x - 1) / 2 plus 1, which is x. And the inverse of f of x is (2x + 1 - 1) / 2 = x. Both come back to x, so the check passes. A quick pipeline: f(4) is 9, and the inverse of 9 is (9 - 1) / 2 = 4. **Example.** Now a rule with a fraction. Find the inverse of f(x) = (x - 4) / 3. Write y = (x - 4) / 3. Swap: x = (y - 4) / 3. Rearrange: multiply by 3 to get 3x = y - 4, then add 4: y = 3x + 4. Check: f of the inverse of x is (3x + 4 - 4) / 3, which is 3x / 3 = x. Notice the pattern. The original subtracts 4, then divides by 3. The inverse multiplies by 3, then adds 4. The operations come back in reverse order, each undone in its turn. **Example.** Finally, the picture. The graph of the inverse is the reflection of the original in the line y = x. Why: swapping x and y trades each point's coordinates, and that is exactly what a mirror in y = x does to a point. The point (0, 1) on the graph of f becomes (1, 0) on the graph of the inverse, and (2, 5) becomes (5, 2). A straight line reflects to a straight line, so the inverse of a linear rule is always linear. Curves reflect too, but their shapes change, so the swap method is the one that keeps the work exact. **Recap.** Find the inverse of a linear rule by swapping x and y and rearranging to make y the subject, check it by composing it with the original to get x back, and remember that the graph reflects in the line y = x. ## Practice 24 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)