Hint
Write both compositions before expanding. Equality for a few inputs does not make the two functions identical.
Worked solution
(a)
\(f(g(x))=2(x^2+2)-1=2x^2+3\). Also \(g(f(x))=(2x-1)^2+2=4x^2-4x+3\).
Answer: \(fg(x)=2x^2+3\); \(gf(x)=4x^2-4x+3\).
(b)
Equating gives \(2x^2+3=4x^2-4x+3\), hence \(2x(x-2)=0\). Both roots 0 and 2 are allowed.
Answer: \(x=0\) or \(x=2\).
(c)
Since \(x^2\ge0\), the minimum of \(2x^2+3\) is 3 at \(\displaystyle x=0\). Every greater value is attained, so the range is \(\displaystyle y\ge 3\).
Answer: \([3,\infty)\).
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) · 3 marks
- 1 mark: Obtain fg=\(\displaystyle 2x^{2}+3\).
- 1 mark: Substitute \(\displaystyle (2x-1)^{2}+2\) for gf.
- 1 mark: Expand gf correctly to \(\displaystyle 4x^{2}-4x+3\).
Alternative methods: Unexpanded but correct expressions earn the relevant substitution criterion; simplification is required for its accuracy criterion.
Common error: Reversing composition order or omitting the cross term in a square.
(b) · 3 marks
- 1 mark: Equate the two compositions.
- 1 mark: Reduce to a correct factored equation.
- 1 mark: Give both roots 0 and 2.
Alternative methods: Any valid quadratic solution method is accepted.
Common error: Dividing by x and losing zero.
(c) · 2 marks
- 1 mark: Identify and justify minimum 3.
- 1 mark: Give the full range \(\displaystyle y\ge 3\).
Alternative methods: A graph or completed-square argument is accepted.
Common error: Finding the range of gf instead of the requested composition.
Total: 8 marks.