O-LevelAdditional MathematicsFunctions4037/21 · May/June 2025 · Q78 Marks

The function f is defined by f(x) = 2e⁻ˣ +3 for x ∈ R. On the axes, sketch the graph of y = f(x) and hence, on the same axes, sketch the graph of y = f⁻¹(x). Show clearly • the positions of any points where your graphs meet the coordinate axes • the positions of any asymptotes.

(a)The function f is defined by f(x) = 2e⁻ˣ +3 for x ∈ R. On the axes, sketch the graph of y = f(x) and hence, on the same axes, sketch the graph of y = f⁻¹(x). Show clearly • the positions of any points where your graphs meet the coordinate axes • the positions of any asymptotes.[4]
(b)The function g is defined by g(x) = 2 - 3/(eˣ+2) for x ≥ 0. Given that g⁻¹ exists, find an expression for g⁻¹(x) and state its domain.[4]

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Mark scheme

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

[B1] for correct shape for f; must have positive y-intercept and appear to tend to an asymptote in the 1st quadrant
[B1] for (0, 5) indicated; must have attempted correct shape for f
[B1] for asymptote at y = 3; must have attempted correct shape for f
[B1] for a correct reflection of their f in the line y = x
y = x does not have to be drawn
Maximum of B3 if not fully correct
[M1] Rearranges to e^x = ... e.g. e^x = (2x-1)/(2-x) or e^x = (1-2x)/(x-2)
[A1] ln(3/(2-x) - 2) oe, isw
[B2] B1 x ≥ 1 oe OR x < 2 oe OR 1 < x ≤ 2
[A1] 1 < x ≤ 2

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