Functions
AI Assistant

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?"

Graph

FAQ

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).

Related Topics

What is a Function?Function CompositionLogarithmic Functions