Hint
Complete the square. If the domain extends beyond the vertex it contains a pair of equal-output inputs.
Worked solution
(a)
\(f(x)=(x-3)^2+4\). It strictly decreases on \(\displaystyle x\le 3\), so \(\displaystyle a=3\) works. For any a>3 choose a sufficiently small t>0 with \(\displaystyle 3+t\le a\). Both 3−t and 3+t are in the domain and have equal outputs, so no larger a works.
Answer: \(a=3\).
(b)
On \(\displaystyle x\le 3\), solve \(\displaystyle y=(x-3)^{2}+4\) using \(\displaystyle x-3=-\sqrt{y-4}\). Thus the inverse is \(\displaystyle 3-\sqrt{x-4}\), with domain \(\displaystyle x\ge 4\).
Answer: \(3-\sqrt{x-4}\), \(x\ge4\).
(c)
\(\displaystyle (x-3)^{2}=4\) gives \(\displaystyle x=1\) or 5. The domain \(\displaystyle x\le 3\) excludes 5, so the only solution is 1.
Answer: \(x=1\) only.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 2 marks
- 1 mark: Locate the vertex at \(\displaystyle x=3\) and identify the one-to-one left branch.
- 1 mark: State \(\displaystyle a=3\) and explain why any larger endpoint repeats an output.
Alternative methods: A precise graph argument using symmetric inputs is accepted.
Common error: Giving a<3 as the greatest value or merely finding the vertex without considering larger a.
(b) · 3 marks
- 1 mark: Choose the negative square-root branch.
- 1 mark: Obtain \(\displaystyle 3-\sqrt{x-4}\).
- 1 mark: State inverse domain \(\displaystyle x\ge 4\).
Alternative methods: An equivalent rearranged formula is accepted.
Common error: Selecting the positive branch despite \(\displaystyle x\le 3\).
(c) · 2 marks
- 1 mark: Obtain candidate roots 1 and 5.
- 1 mark: Reject 5 using the domain and give 1 only.
Alternative methods: Evaluating \(\displaystyle f^{−1}(8)=1\) with its branch justified is accepted.
Common error: Keeping the root on the other branch.
Total: 7 marks.