O-LevelAdditional MathematicsFunctions4037/12 · Oct/Nov 2022 · Q810 Marks

A function f(x) is such that f(x) = ln(2x+3)+ln4, for x > a, where a is a constant.

(a)Write down the least possible value of a.[1]
(b)Using your value of a, write down the range of f.[1]
(c)Using your value of a, find f⁻¹(x), stating its range.[4]
(d)On the axes below, sketch the graphs of y = f(x) and y = f⁻¹(x), stating the exact intercepts of each graph with the coordinate axes. Label each of your graphs.[4]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[1] (a) -1.5
[1] (b) y ∈ R, or R, or -∞ < f(x) < ∞, or f(x) ∈ R
[1] (c) f(x) = ln(8x+12) or ln(4(2x+3)) (may be implied)
[1] (c) For attempt to find the inverse (e.g. e^y = 8x+12), allow one sign error
[1] (c) f⁻¹(x) = (eˣ-12)/8 or equivalent (allow y = ...)
[1] (c) Range: f⁻¹(x) > -1.5 or y > -1.5 (must be correct notation, follow through on their (a))
[1] (d) For correct shape of y = f(x) in quadrants 1, 2 and 3, with asymptotic behaviour
[1] (d) For intercepts ln12 and -11/8 or -1.375 in correct position for f(x), must have a correct shape.
[1] (d) For correct shape of y = f⁻¹(x) in quadrants 1, 3 and 4, with asymptotic behaviour
[1] (d) For intercepts ln12 and -11/8 or -1.375 in correct position for f⁻¹(x), must have a correct shape and intersect at least once with y=f(x)

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