Inverse functions

Objective. Find and verify inverses of one-to-one functions.

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Learn: the key idea

An inverse function f⁻¹(x) reverses the action of f(x), undoing it so that f⁻¹(f(x))=x, and it only exists properly when f is one-to-one over its stated domain. To find f⁻¹, write y=f(x), swap x and y, then rearrange to make y the subject again. For f(x)=2x+3, swapping gives x=2y+3, so y=(x−3)/2, meaning f⁻¹(x)=(x−3)/2. The graph of f⁻¹ is the reflection of f in the line y=x, and the domain of f⁻¹ equals the range of f.

Worked example

Find the inverse function f⁻¹(x) for f(x) = (x+1)/4.

1 $Write y = (x+1)/4$
2 $Swap x and y to get x = (y+1)/4$
3 $Multiply both sides by 4: 4x = y+1$
4 $Rearrange for y to get f⁻¹(x) = 4x - 1$

Answer: f⁻¹(x) = 4x − 1

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Common mistake. Students find the reciprocal 1/f(x) instead of the true inverse; for f(x)=x+5, writing f⁻¹(x)=1/(x+5) is wrong, since the correct inverse undoes addition by subtracting 5, giving f⁻¹(x)=x−5.

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