Choosing the branch of an inverse

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

Suggested marks guide how much working to show. The site does not automatically award examination marks for proofs, diagrams or methods. Use scaffolded fluency practice when you need a hint first.

Level 2 / FSMQ. Course filters use selected objective mappings; none of these tasks establishes complete paper coverage.

Before you start

Inverse functions. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

$f(x)=(x-2)^2-1$ , with domain $xgeq2$ .

Part a

Find f⁻¹, including its domain and range.

Part b

Find f⁻¹(8).

Part c

Explain why the domain restriction matters.

Model solution and review criteria

Part a

f⁻¹(x) = 2 + √(x − (-1)). Its domain is x ≥ -1, and range is x ≥ 2.

Interchange input and output.
Select the square-root sign using the given domain.
State the inverse domain and range.

Part b

The square root is 3; the selected branch gives 5.

Use the branch found in part (a).

Part c

Without the restriction, inputs 1 and 3 both have output 0; an inverse function cannot select a unique input.

Give two distinct inputs with the same output.
Explain uniqueness of the inverse.

Compare your own reasoning. No automatic examination marks are awarded.