Hint
The exponential factor is strictly positive. The exponential tends to zero to the left.
Worked solution
(a)
Every real input gives a positive exponential term. Thus the output is strictly above D, approaching but never reaching that height.
Answer: \(\mathbb{R}\); \(\left(9, \infty\right)\); \(y=9\).
(b)
Reverse the linear shift and scale, take \(\displaystyle \log _{2}\), divide by B and add h. Exchange the original domain and range.
Answer: \(\displaystyle \frac{\log{\left(\frac{x}{2} - \frac{9}{2} \right)}}{3 \log{\left(2 \right)}} + 4\); domain \(\left(9, \infty\right)\); range \(\mathbb{R}\).
(c)
The exponent is one.
\(13\)
Answer: \(13\)
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 positivity of the exponential to state the domain, range and limiting horizontal line.
- 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 horizontal asymptote is excluded from the range.
(b) · 2 marks
- 1 mark: Isolate the exponential, take logarithms and exchange domain and range.
- 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: Undo the operations in reverse order.
(c) · 2 marks
- 1 mark: The exponent is one.
- 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: A horizontal asymptote is excluded from the range.
Total: 6 marks.