O-LevelAdditional MathematicsFunctions4037/12 · May/June 2023 · Q810 Marks

It is given that f(x) = 2ln(3x-4) for x > a.

(a)Write down the least possible value of a.[1]
(b)Write down the range of f.[1]
(c)It is given that the equation f(x) = f⁻¹(x) has two solutions. (You do not need to solve this equation). Using your answer to part (a), sketch the graphs of y = f(x) and y = f⁻¹(x) on the axes below, stating the coordinates of the points where the graphs meet the axes.[4]
(d)(i)It is given that g(x) = 2x-3 for x ≥ 3. Find an expression for g(g(x)).[1]
(d)(ii)Hence solve the equation fg(g(x)) = 4 giving your answer in exact form.[3]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[B1] Allow a recurring decimal Must not be an inequality in terms of a (a=) 4/3 or 1.3
[B1] Allow y or f(x) but not x. f ∈ R or -∞ < f < ∞ or R
[B1] for a correct shape for y = f(x) in quadrants 1 and 4
[B1] for (5/3, 0), must have a correct shape in either quadrant 1 or quadrant 4
[B1] for y = f⁻¹(x), must be a correct shape in quadrants 1 and 2 and intersect twice.
[B1] for (0, 5/3), must have a reasonable shape for y = f⁻¹(x) in either the first quadrant or the second quadrant
[B1] Must be simplified g(g(x)) = 4x-9
[M1] allow unsimplified, using their answer to (i) fg(g(x)) = 2ln(12x-31)
[Dep M1] for correct order of operations to solve their equation, to get as far as x = ... Implied by decimal answer of awrt 3.2
[A1] Must be exact form. x = (e² + 31) / 12

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.