Hint
The right-hand branch \(\displaystyle x\ge h\) is specified. The square is nonnegative.
Worked solution
(a)
The minimum is k at \(\displaystyle x=h\); on the right-hand branch the rule is strictly increasing.
Answer: \(\left[5, \infty\right)\)
(b)
\(\displaystyle x = \sqrt{\frac{x}{3} - \frac{5}{3}} + 9\) after interchanging input and output; the new input must be at least k.
Answer: \(\displaystyle \sqrt{\frac{x}{3} - \frac{5}{3}} + 9\), \(\left[5, \infty\right)\)
(c)
\(53 = 53\)
Answer: \(13\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Use nonnegativity of the square and strict increase on the restricted branch.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A ± sign is not an inverse function.
(b) · 2 marks
- 1 mark: Isolate the square, select the permitted branch and exchange input and output.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not omit the inverse domain.
(c) · 2 marks
- 1 mark: Evaluate the inverse on its valid domain and substitute the result into f.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A ± sign is not an inverse function.
Total: 6 marks.