Domains, Ranges and Inverse Functions

≈ 25 min

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Domains, Ranges and Inverse Functions

The inverse interchanges domain and range and reflects the graph in $y=x$ . Restrict a non-one-to-one function before inverting it; the chosen restriction determines which inverse branch is obtained.

Worked reasoning

Write $y=(x-2)^2$ and swap variables.
$x=(y-2)^2$ ; the restriction selects $y-2\ge0$ .
$f^{-1}(x)=2+\sqrt x$ .

Key idea

A valid inverse function requires a one-to-one original domain.

For $f(x)=(x-2)^2$ restricted to $x\ge2$ , find $f^{-1}(x)$ .

Which statement best captures the central mathematical idea in Domains, Ranges and Inverse Functions?

A valid inverse function requires a one-to-one original domain.

Restricting a domain changes only notation and never changes the inverse branch.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Domains, Ranges and Inverse Functions, which move is most reliable?

State the original domain and range and apply the horizontal-line test.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Domains, Ranges and Inverse Functions?

Restricting a domain changes only notation and never changes the inverse branch.

A valid inverse function requires a one-to-one original domain.

State the original domain and range and apply the horizontal-line test.

Compose in both directions on the relevant domains.

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