A-LevelMathematicsFunctions9709/12 · Oct/Nov 2025 · Q98 Marks

The function f is defined by f(x) = 4/(3x-6)² + 1/(3x-6)³ for x > 2. The function g is defined by g(x) = 4x-3 for x > a.

(a)Find an expression for f'(x) and hence determine whether f is an increasing function, a decreasing function or neither.[4]
(b)State whether f⁻¹ exists. Give a reason for your answer.[1]
(c)Find the range of g in terms of the constant a.[1]
(d)Find the set of values of a for which the composite function fg exists.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

[B3] f'(x) = -24(3x-6)^(-3) - 9(3x-6)^(-4) (B1 for each term differentiated correctly, B1 for chain rule factor of 3 applied to both terms. Penalise missing x-terms in powers only if present in other terms).
[B1] States 'decreasing' with correct reasoning (f'(x) is always negative for x>2).
[DB1] f⁻¹ exists because f is a one-to-one function (or decreasing function / passes the horizontal line test).
[B1] Range of g is g(x) > 4a - 3 (or (4a-3, ∞)).
[M1] Sets the range of g (g(x) > 4a-3) to be within the domain of f (x > 2), i.e., 4a-3 > 2 (or 3(4a-3)-6 > 0).
[A1] a > 5/4 or a ≥ 5/4 (or (5/4, ∞)).

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