2024 HSC Mathematics Extension 1
Question 14(b):
Inverse function
For what values of the constant would the function have an inverse?
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
- is defined and differentiable on all . It has an inverse iff it is strictly monotone, i.e. for all or for all .
- Differentiate: derivative of is , plus .
- Combine: .
- Denominator always positive. For always: need and (both coefficients of the quadratic in non-negative), i.e. .
- For always: need and , i.e. and , impossible.
- Hence has an inverse precisely when .
- NESA 14(b): 3 marks correct; 2 marks for completing one of the two cases; 1 mark for recognising the need for monotone derivative sign.
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
For what values of does have an inverse?
Question 2
Show that has the same sign as for , and deduce the values of for which on .
Question 3
For , compute when and when , and conclude whether an inverse exists in each case.