Hint
The logarithm argument must be positive. The argument approaches zero at \(\displaystyle x=h\).
Worked solution
(a)
Require a(x−h)>0 with a>0.
Answer: \(\left(9, \infty\right)\); \(x = 9\)
(b)
Isolate the logarithm, exponentiate, solve for the original input and exchange the variables.
Answer: \(\displaystyle e^{\frac{x}{3} - \frac{5}{3}} + 9\); domain \(\mathbb{R}\)
(c)
Solve \(\displaystyle a(x-h)=1\).
\(10\)
Answer: \(10\)
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: Require a positive logarithm argument and locate the excluded boundary.
- 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: The asymptote is determined by the argument zero, not the vertical translation.
(b) · 2 marks
- 1 mark: Reverse the logarithm using an exponential and state all permitted inverse inputs.
- 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: Interchange domain and range for the inverse.
(c) · 2 marks
- 1 mark: Solve \(\displaystyle a(x-h)=1\).
- 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: The asymptote is determined by the argument zero, not the vertical translation.
Total: 6 marks.