Relations and Inverse Functions

An inverse function undoes whatever the original function did, sending every output back to the input that produced it. Not every function has one — whether it does comes down to a single graphical check.

Concept The idea of an inverse

If squares a number, must take a square root to undo it. If doubles a number, must halve it. Applying a function and then its inverse always returns you to where you started.

The notation

is read "f inverse" — the

is a label, not an exponent. It does not mean

Example Checking an inverse pair

Verify that and really are inverses of each other.

Start with

Feed the output back through the inverse:

⟹ we land back on 4 — the original input

This round trip — apply , then apply , and recover the starting value — is exactly what it means for two functions to be inverses.

Concept The graphical relationship

The graph of

is the reflection of the graph of

across the line

Every point

corresponds to the point

— the coordinates simply swap.

Concept The horizontal line test

A relation is a function only if it passes the vertical line test — every vertical line meets the graph at most once, so every input gives exactly one output. A function has an inverse only if it also passes the horizontal line test — every horizontal line meets the graph at most once, so every output came from exactly one input. This second property is called one-to-one.

If some horizontal line crosses the graph twice, then two different inputs share the same output, and there is no way to reverse the process unambiguously — the "inverse" would not know which input to send that output back to.

Note Five functions tested

Function	Vertical line test	Horizontal line test	Has an inverse?
y = 2x	Pass	Pass	Yes
y = x³	Pass	Pass	Yes
y = x²	Pass	Fail	No
y = \|x\|	Pass	Fail	No
y = 5	Pass	Fail	No

Notice that fails only because both and give . Restricting the domain to removes the repeated output, and on that restricted piece the function does have an inverse: .

Summary

An inverse function undoes the original: f⁻¹(f(x)) = x.
f⁻¹ exists only if f passes the horizontal line test — it must be one-to-one.
The graph of f⁻¹ is the reflection of the graph of f across y = x; point (a, b) becomes (b, a).
A function that fails the horizontal line test can still gain an inverse once its domain is restricted to remove repeated outputs.