Hint
Use the stated units and precision. Apply the inner rule first and substitute its whole output into the outer rule before simplifying.
Worked solution
(a)
\(fg(x)=5 \left(x - 1\right)^{2} + 5\) expands to \(5 x^{2} - 10 x + 10\).
Answer: \(5 x^{2} - 10 x + 10\)
(b)
\(gf(x)=\left(5 x + 4\right)^{2}\) expands to \(25 x^{2} + 40 x + 16\).
Answer: \(25 x^{2} + 40 x + 16\)
(c)
The values at this t differ. Independently, the \(\displaystyle x^{2}\) coefficients of the complete rules are 5 and 25, which are unequal.
Answer: \(fg(t)=130,\quad gf(t)=1156\). They are different functions.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 2 marks
- 1 mark: Substitute the whole expression g(x) into f and expand consistently.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
(b) · 2 marks
- 1 mark: Substitute the whole expression f(x) into g, including its square.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Check the answer against the original conditions.
(c) · 3 marks
- 1 mark: Evaluate both orders at the same input.
- 1 mark: Use the different complete expressions or a differing value to establish that the functions are not identical.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
Total: 7 marks.