Hint
On this branch x−2 is nonnegative. That determines the square-root sign.
Worked solution
(a)
As x increases from 2, the nonnegative quantity x−2 increases, so its square strictly increases. Thus no output is repeated. The minimum is −3 at \(\displaystyle x=2\) and there is no upper bound.
Answer: Strictly increasing for \(x\ge2\); range \([-3,\infty)\).
(b)
From \(\displaystyle y=(x-2)^{2}-3\), take \(\displaystyle x-2=\sqrt{y+3}\) because \(\displaystyle x\ge 2\). Hence the inverse is \(\displaystyle 2+\sqrt{x+3}\). Its domain is the original range and its range is the original domain.
Answer: \(2+\sqrt{x+3}\); domain \(x\ge-3\), range \(y\ge2\).
(c)
\(\displaystyle 2+\sqrt{x+3}=6\) gives \(\displaystyle \sqrt{x+3}=4\), so \(\displaystyle x=13\). It is in the inverse domain. Equivalently \(\displaystyle x=f(6)=13\).
Answer: \(x=13\).
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: Justify one-to-one behaviour on the stated right-hand branch.
- 1 mark: Give range \(\displaystyle y\ge -3\).
Alternative methods: A correct monotonic graph or derivative argument is accepted; a zero derivative at the endpoint does not prevent strict increase.
Common error: Claiming that every quadratic is one-to-one.
(b) · 3 marks
- 1 mark: Choose and obtain the positive-branch inverse \(\displaystyle 2+\sqrt{x+3}\).
- 1 mark: State inverse domain \(\displaystyle x\ge -3\).
- 1 mark: State inverse range \(\displaystyle y\ge 2\).
Alternative methods: Rearrangement with an explicit branch check is accepted.
Common error: Giving \(\displaystyle \pm\sqrt{\cdot}\) as if the inverse could have two outputs.
(c) · 2 marks
- 1 mark: Use a valid inverse equation or \(\displaystyle x=f(6)\).
- 1 mark: Obtain the permitted solution 13.
Alternative methods: Direct evaluation of f(6) is accepted.
Common error: Squaring 6 before subtracting 2.
Total: 7 marks.