iGCSE Mathematics (0580) :E2.9 Use function notation.iGCSE Style Questions Paper 4

Question

\(f\left ( x \right )=7x-2\) \(g\left ( x \right )=x^{2}+1\) \(h\left ( x \right )=3^{x}\)

(a) Find gh(2).

(b) Find f ^{– 1}(x).

(c) \(gg\left ( x \right )=ax^{4}+bx^{2}+c\)

Find the values of a, b and c.

(d) Find x when hf(x) = 81.

▶️ Answer/Explanation

Solution

(a) Ans: 82

First compute \(h(2) = 3^2 = 9\). Then, \(g(h(2)) = g(9) = 9^2 + 1 = 82\).

(b) Ans: \(\frac{x+2}{7}\)

Let \(y = 7x – 2\). Swap \(x\) and \(y\) and solve for \(y\): \(x = 7y – 2 \Rightarrow y = \frac{x+2}{7}\).

(c) Ans: a = 1, b = 2, c = 2

Compute \(g(g(x)) = (x^2 + 1)^2 + 1 = x^4 + 2x^2 + 1 + 1 = x^4 + 2x^2 + 2\). Thus, \(a = 1\), \(b = 2\), \(c = 2\).

(d) Ans: \(\frac{6}{7}\)

Given \(h(f(x)) = 3^{7x-2} = 81 = 3^4\). So, \(7x – 2 = 4 \Rightarrow x = \frac{6}{7}\).

Question

The diagram shows the speed-time graph for the first 180 seconds of a train journey.

(a) Find the acceleration, in m/s², of the train during the first 50 seconds.

(b) After 180 seconds, the train decelerates at a constant rate of 1944 km/h².

Show that the train decelerates for 60 seconds until it stops.

(c) Complete the speed-time graph.

(d) Calculate the average speed of the train for the whole journey.

▶️ Answer/Explanation

Solution

(a) 0.18 m/s²

Acceleration is change in speed over time. From the graph, speed increases from 0 to 9 m/s in 50 seconds.

(b) Convert 1944 km/h² to m/s²: 1944 × (1000/3600²) = 0.15 m/s². Time to stop = 9 m/s ÷ 0.15 m/s² = 60 seconds.

(c)

The graph should show constant speed until 180s, then linear deceleration to 0 at 240s.

(d) 6.94 m/s

Total distance is area under graph: (½×50×9) + (130×9) + (½×60×9) = 1665m. Average speed = 1665m ÷ 240s.

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