Inverse function — Exponential plus 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 all real , so is strictly increasing, hence one-to-one, so is 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: Full (i)–(ii): injectivity then inverse derivative at the known point.

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