Inverse function — k for invertibility

Question

For what values of does have an inverse?

This practice question was inspired by Question 14(b) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: replace the function value by y, swap x and y, then solve for y..
Finish the calculation, then check that the result meets the question’s conditions.Need for all , i.e. .
State the final answer clearly in the form the question requested.Answer: .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Denominator always positive; only numerator sign matters.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.