2022 HSC Mathematics Extension 1
Question 13(c):
Inverse function
The function is defined by for all real . Let be defined on by .
Is the inverse of ? Justify your answer.
How to recognise this question
Question type: inverse functions
- The command and notation point to inverse functions.
- The requested response is a explanation.
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
- For to be the inverse of , we would need for all in the domain of , and on .
- While on , the composition equals only for .
- Counterexample: . Then , so .
- Also is not one-to-one on , so it has no inverse on its full domain.
- Therefore is not the inverse of (as defined on all reals).
- NESA: 2 marks; 1 mark for a value where .
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
If instead on , is its inverse?
Question 2
Show .
Question 3
Is the inverse of on all of ? Justify.