Inverse Functions
An inverse undoes a function and swaps inputs with outputs.
About this topic
An inverse function f⁻¹ reverses f, so f⁻¹(f(x)) = x. To find it, swap x and y in y = f(x) and solve for y. The graph of the inverse is the reflection of the original across the line y = x.
Only one-to-one functions have inverses, which the horizontal line test checks. Domains and ranges swap. Examples include logarithms undoing exponentials and the formula for converting Fahrenheit back to Celsius.
Worked example: find the inverse of f(x) = 3x − 6. Write y = 3x − 6, swap to get x = 3y − 6, solve to get y = (x + 6)/3. Check: f(f⁻¹(x)) = 3((x + 6)/3) − 6 = x. The temperature rule F = 1.8C + 32 has the inverse C = (F − 32)/1.8, which is what a weather app uses when switching units. The graphs of f and its inverse mirror each other across the line y = x, so a point (2, 0) on f gives (0, 2) on the inverse.
Inverses are essential for solving equations: solving 2ˣ = 20 means applying the inverse, a logarithm. A frequent error is mistaking f⁻¹(x) for 1/f(x); the exponent means inverse, not reciprocal. For functions like x², restrict the domain to x at least 0 so the function becomes one-to-one and the inverse, the square root, is well defined.
Key ideas
- Swap x and y then solve for y
- Graphs reflect over y = x
- Only one-to-one functions have inverses
Quick check
Use cases: where Inverse Functions helps
- Homework: review the key idea behind Inverse Functions before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Inverse Functions without notes.
- Visual learning: play with the interactive lab to see how Inverse Functions changes when you adjust its inputs.
- Teaching a friend: explain Inverse Functions in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Inverse Functions?
An inverse undoes a function and swaps inputs with outputs.
What will I learn in Inverse Functions?
Swap x and y then solve for y. Graphs reflect over y = x. Only one-to-one functions have inverses.
Is Inverse Functions part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Inverse Functions lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Inverse Functions?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.