Inverse Functions

Find the function that undoes another — reflect over y = x

An inverse function reverses what the original function does. If f(x) = 2x + 3 turns 2 into 7, then f⁻¹(x) turns 7 back into 2.

To find the inverse: swap x and y, then solve for y. The graph of f⁻¹ is the mirror image of f reflected over the line y = x.

Not every function has an inverse — it must pass the horizontal line test. Ask the AI anything — try "Find the inverse of f(x) = 3x − 1" or "Why doesn't x² have an inverse?"

Related Topics

What is a Function?Function CompositionLogarithmic Functions

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
What is an inverse function?
An inverse function f⁻¹(x) "undoes" the original function f(x). If f takes input a to output b, then f⁻¹ takes b back to a. Formally, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
How do I find an inverse function?
Write y = f(x), swap x and y to get x = f(y), then solve for y. For example: y = 2x + 3 becomes x = 2y + 3, then y = (x − 3)/2. So f⁻¹(x) = (x − 3)/2.
Why is the inverse a reflection over y = x?
When you swap x and y in every point (a, b) on f, you get (b, a) on f⁻¹. The transformation (a, b) → (b, a) is exactly a reflection across the line y = x.
When does an inverse not exist?
A function has an inverse only if it is one-to-one — each output comes from exactly one input. The horizontal line test checks this: if any horizontal line crosses the graph more than once, the function has no inverse (unless you restrict the domain).
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.