O-LevelAdditional MathematicsLogarithmic and exponential functions4037/12 · Oct/Nov 2024 · Q413 Marks

The function f is such that f(x) = 4ln(3x-2), for x > a, where a is as small as possible.

(a)(i)Write down the value of a.[1]
(a)(ii)Write down the range of f.[1]
(a)(iii)Find f⁻¹(x), stating its domain and range.[4]
(a)(iv)On the axes sketch the graphs of y = f(x) and y = f⁻¹(x), stating the intercepts with the axes.[4]
(b)Given that g(x) = (2x+1)³+4, for x > 0, solve the equation gg(x) = 9.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[B1] 2/3
[B1] R oe
[M1] 3y-2 = e^(x/4) or 3x-2 = e^(y/4) (For valid attempt to reach this stage)
[A1] f⁻¹(x) = (1/3)e^(x/4) + 2/3 (Must be using correct notation)
[B2] B1 for each, must be using the correct notation. Domain x ∈ R, Range f⁻¹ > 2/3
[B1] for the shape of y = f (x) in the first and fourth quadrants only
[B1] dep on previous B1 for (1, 0)
[B1] for a correct shape for f⁻¹ (x), or FT on their y = f (x) with correct shape in first quadrant for symmetry about y = x soi
[B1] dep on previous B1, for (0, 1) and at least one point of intersection with y = f (x) correct in the first quadrant
[B1] (2{(2x+1)³+4}+1)³ + 4
[B1] (2x+1)³ = 8
[B1] x = 31.5 oe

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