Inverse function — Gentler log-linear

Question

Let for .

(i) Explain why the inverse of is a function.

(ii) Let . By considering , evaluate .

This practice question was inspired by Question 14(a) in the 2023 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.(i) for , so is strictly increasing on , hence invertible as a function.
Finish the calculation, then check that the result meets the question’s conditions.(ii) , so .
State the final answer clearly in the form the question requested..

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

Verification: Both parts required: monotonicity argument, then via .

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