Inverse function — Boundary k checks

Question

For , compute when and when , and conclude whether an inverse exists in each case.

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.: — inverse exists.
Finish the calculation, then check that the result meets the question’s conditions.: with isolated zero at — still non-decreasing and bijective onto range; inverse exists.
State the final answer clearly in the form the question requested.Answer: inverse exists for both and (endpoints of ).

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

Verification: Endpoints of the NESA interval are included.

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