Hint
Use the stated units and precision. Use the stated input interval, evaluate the linear rule or reverse its operations for an inverse input.
Worked solution
(a)
The positive gradient 5 makes f strictly increasing. Its endpoint outputs are 5 and 35, and both are attained.
Answer: \(\left[5, 35\right]\); f is one-to-one on its stated domain.
(b)
Solve \(y=5 x + 10\) for x, interchange the variables, and exchange the original domain and range.
Answer: \(\displaystyle f^{-1}(x)=\frac{x}{5} - 2\); domain \(\left[5, 35\right]\), range \(\left[-1, 5\right]\).
(c)
\(f(0)=10\) verifies the reverse calculation. The input 0 lies in the original domain.
Answer: \(0\)
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) · 3 marks
- 1 mark: Use monotonicity and both included domain endpoints to identify the range.
- 1 mark: Explain why distinct allowed inputs have distinct outputs.
- 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) · 3 marks
- 1 mark: Reverse the linear rule and interchange input and output.
- 1 mark: Give both exchanged sets with their correct included endpoints.
- 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) · 2 marks
- 1 mark: Evaluate the inverse on its valid domain and verify the original-function round trip.
- 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: 8 marks.