2023 HSC Mathematics Extension 1
Question 14(a):
Inverse function
Let for .
(i) Explain why the inverse of is a function.
(ii) Let . By considering the value of , or otherwise, evaluate .
How to recognise this question
Question type: inverse functions
- The command and notation point to inverse functions.
- The word “hence” means the later part is intended to reuse the earlier result.
How to handle it: replace the function value by y, swap x and y, then solve for y.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- (i) . For , , so .
- Thus is strictly increasing on , hence one-to-one, so is a function.
- NESA (i): 1 mark for a correct explanation.
- (ii) , so .
- Inverse derivative: , hence .
- NESA (ii): 2 marks correct; 1 for finding formula or .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Let for .
(i) Explain why the inverse of is a function.
(ii) Let . By considering , evaluate .
Question 2
Let for .
(i) Explain why is a function.
(ii) Let . By considering , evaluate .
Question 3
Let for .
(i) Explain why the inverse of is a function.
(ii) Let . By considering , evaluate .