O-LevelAdditional MathematicsFunctions4037/22 · May/June 2015 · Q108 Marks

The function f is defined by f:x↔|sinx| for 0° ≤x≤ 360°. On the axes below, sketch the graph of y = f(x). [Figure 10.1]

(a)The function f is defined by f:x↔|sinx| for 0° ≤x≤ 360°. On the axes below, sketch the graph of y = f(x). [Figure 10.1][2]
(b)(i)The functions g and hg are defined, for x ≥ 1, by g(x) = ln (4x - 3), hg(x) = x. Show that h(x) = (eˣ + 3) / 4.[2]
(b)(ii)[Figure 10.2] The diagram shows the graph of y = g(x). Given that g and h are inverse functions, sketch, on the same diagram, the graph of y = h(x). Give the coordinates of any point where your graph meets the coordinate axes.[2]
(b)(iii)State the domain of h.[1]
(b)(iv)State the range of h.[1]

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Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

[B2,1,0] correct sinusoidal/reflected sinusoidal shape, all above x-axis with intent to have all maximum points of equal height;
2 maximum points of intended equal height only over 0 to 360;
all max points clearly at y = 1;
cusp at 180
[M1] [hg(x) =] e^(ln(4x-3)) + 3 / 4 Alternative method: y = ln(4x-3) and change of subject to x oe,
[A1] fully correct and completion to [hg(x) =] x fully correct and comment that h(x) = g⁻¹(x) oe
[B2,1,0] correct shape; 1 marked on the y-axis or (0, 1) stated close by; curve with positive gradient in first quadrant only
[B1] x ≥ 0 or [0, ∞) not domain ≥ 0
[B1] y ≥ 1 or [1, ∞) or h(x) ≥ 1, h ≥ 1 etc.

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