Relations and Inverse Functions

Relations and Inverse Functions — Moosa Academy

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

f^{-1}(f(x)) = x

If $f$ squares a number, $f^{-1}$ must take a square root to undo it. If $f$ doubles a number, $f^{-1}$ must halve it. Applying a function and then its inverse always returns you to where you started.

The notation

f^{-1}

is read "f inverse" — the

-1

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

\dfrac{1}{f(x)}

Example Checking an inverse pair

Verify that $f(x) = 2x$ and $f^{-1}(x) = \dfrac{x}{2}$ really are inverses of each other.

Start with

x = 4

f(4) = 2(4) = 8

Feed the output back through the inverse:

f^{-1}(8) = \dfrac{8}{2} = 4

⟹ we land back on 4 — the original input

This round trip — apply $f$ , then apply $f^{-1}$ , and recover the starting value — is exactly what it means for two functions to be inverses.

Concept The graphical relationship

The graph of

f^{-1}

is the reflection of the graph of

f

across the line

y = x

Every point

(a,\ b)

f

corresponds to the point

(b,\ a)

f^{-1}

— 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 $y = x^2$ fails only because both $x = 2$ and $x = -2$ give $y = 4$ . Restricting the domain to $x \geq 0$ removes the repeated output, and on that restricted piece the function does have an inverse: $\sqrt{x}$ .

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.