Hint
State which notation denotes composition. Evaluate g before f.
Worked solution
(a)
\(4 \left(x - 4\right)^{2} + 12\) is obtained by substituting the entire expression g(x) into f.
Answer: \(4 x^{2} - 32 x + 76\)
(b)
\(\left(4 x + 8\right)^{2}\) follows by squaring the full linear expression f(x)−h.
Answer: \(16 x^{2} + 64 x + 64\)
(c)
\(\left(x - 4\right)^{2} = 16\) gives \(\displaystyle x-h=\pm d\); only \(\displaystyle x-h=d\) is in the stated domain.
Answer: \(8\)
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: Substitute the whole inner rule g into f and simplify.
- 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: Composition is not multiplication.
(b) · 2 marks
- 1 mark: Substitute the whole inner rule f into g and simplify.
- 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 assume fg=gf.
(c) · 2 marks
- 1 mark: Solve the squared equation and retain only the branch \(\displaystyle x\ge h\).
- 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: Composition is not multiplication.
Total: 6 marks.