Hint
Use the domain given in the question. Substitute the lower endpoint into the rule.
Worked solution
(a)
The gradient 4 is positive, so the endpoint images are 21 and 45.
Answer: \(\left[21, 45\right]\)
(b)
The nonzero gradient makes f strictly increasing. Solve \(\displaystyle y=4x+13\) for x, then interchange input and output; the inverse domain is the range just found.
Answer: \(\displaystyle \frac{x}{4} - \frac{13}{4}\)
(c)
Solve the linear equation for its input.
\(3\)
Answer: \(3\)
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 monotonicity and evaluate both included domain endpoints.
- 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 confuse a set of inputs with a set of outputs.
(b) · 2 marks
- 1 mark: Justify one-to-one behaviour and reverse the linear rule.
- 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 function can be many-to-one.
(c) · 2 marks
- 1 mark: Solve the linear equation for its input.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not confuse a set of inputs with a set of outputs.
Total: 6 marks.