Right-hand branch
For f(x) = (x − 2)² + 1, x ≥ 2, find f⁻¹(10).
- Set (x − 2)² = 9.
- Because x ≥ 2, choose x − 2 = 3.
Answer: 5
Inverse functions and domain restrictions
Find the inverse of a quadratic on each side of its turning point.
Before you start
You’ll need Domain and range, Rearranging formulae. Follow a link if you want to revise it first.
Restricting the domain
An inverse undoes a function on its stated domain. A reciprocal, 1/f(x), is a different operation.
A quadratic needs a restricted domain to be one-to-one. The restriction determines which square-root branch is allowed.
The range of the original function becomes the domain of its inverse. Include endpoints when they are attained.
The same quadratic has different inverses on x ≥ 2 and x ≤ 2. The original domain determines the sign.
The two square-root branches
Left-hand branch
For f(x) = (x − 2)² + 1, x ≤ 2, find f⁻¹(10).
- Again (x − 2)² = 9.
- Because x ≤ 2, choose x − 2 = −3.
Answer: −1
The ± sign gives both possible inputs before the domain restriction; an inverse function needs the permitted one.
Practice questions
The first link uses smaller values. Move on when you can answer without referring to the examples. The difficulty settings aren’t tied to exam grades.
- Evaluate an inverse on its permitted branch — smaller values
- Find a quadratic inverse with its domain
Practice is filtered to your selected course. Change course or options when needed.
Check your understanding
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What is the domain of each inverse in these examples, and why?