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Inverse Functions

Undoing — and the mirror across y = x.

Interactive graph

Interactive Lab

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Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualAn inverse undoes the original. If f sends 2 → 8, then f⁻¹ sends 8 → 2.

Algebra $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$ .

f^{-1}\!\big(f(x)\big) = x

VisualThe inverse is the original graph reflected across the line y = x. Watch the two moving points swap coordinates.

AlgebraIf $(a,b)$ is on $f$ , then $(b,a)$ is on $f^{-1}$ .

VisualSwap x and y, then solve for y.

AlgebraStep 1: replace $f(x)$ with $y$ . Step 2: swap $x \leftrightarrow y$ . Step 3: solve for $y$ .

y = f(x) \xrightarrow{\text{swap}} x = f(y) \xrightarrow{\text{solve}} y = f^{-1}(x)

VisualOnly one-to-one functions have inverses. If a horizontal line cuts the graph twice, no inverse (without restricting domain).

AlgebraPass the horizontal line test → one-to-one → invertible.

Inverse identity

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

f⁻¹ undoes f

Reflection

(a,b) \in f \Rightarrow (b,a) \in f^{-1}

Across y = x

Domain/range swap

\text{dom}(f^{-1}) = \text{ran}(f)

Inputs and outputs exchange

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easyinverses

f(x)=2x

, find

f^{-1}(x)

easyinverses

f(3) = 7

, what is

f^{-1}(7)

mediuminverses

Find the inverse of

f(x)=3x-5

mediuminverses

Does

f(x)=x^2

have an inverse over its entire domain?

hardinverses

Find the inverse of

f(x)=\dfrac{2x+1}{x-3}

x \neq 3