2021 HSC Mathematics Extension 1
Question 14(e):
Inverse function
The polynomial passes through the point .
Find the gradient of the tangent to at the point where .
How to recognise this question
Question type: inverse functions
- The command and notation point to inverse functions.
- The requested response is a worked solution.
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
- Since , it follows that .
- Product rule: .
- Inverse derivative: .
- Thus .
- At : .
- , so .
- .
- NESA: 2 marks full; 1 for an expression for the derivative of (or equivalent).
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 2 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
With , note so . For , explain why is not the exam target, and compute as above.
Question 2
For , find .
Question 3
If where and , , compute .