UCSI International School
Proportion and Functions
27.3.17
Direct and Inverse Proportion
1.
A spray can is used to paint a wall.
The thickness of the paint on the wall is t. The distance of the spray can from the wall is d.
t is inversely proportional to the square of d.
t = 0.2 when d = 8.
Find t when d = 10.
Answer t = ………….………………
[3]
2.
The quantity p varies inversely as the square of (q + 2).
p = 5 when q = 3.
Find p when q = 8.
Answer p = ………..……………
[3]
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3.
The quantity y varies as the cube of (x + 2).
y = 32 when x = 0.
Find y when x = 1.
Answer y = ………….…………
[3]
4.
The wavelength, w, of a radio signal is inversely proportional to its frequency, f. When f = 200,
w = 1500.
(a)
Find an equation connecting f and w.
Answer (a) ……….………………….……
[2]
(b)
Find the value of f when w = 600.
Answer (b) f = ……….………………….…..
[1]
5.
The force of attraction (F) between two objects is inversely proportional to the square of the
distance (d) between them.
When d = 4, F = 30.
Calculate F when d = 8.
Answer F = …………………….…………
[3]
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6.
The length, y, of a solid is inversely proportional to the square of its height, x.
(a)
Write down a general equation for x and y.
Show that when x = 5 and y = 4.8 the equation becomes x2 y = 120.
[2]
(b)
Find y when x = 2.
[1]
(c)
Find x when y = 10
[2]
(d)
Find x when y = x.
[2]
(e)
Describe exactly what happens to y when x is doubled.
[2]
(f)
Describe exactly what happens to x when y is decreased by 36%.
[2]
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(g)
Make x the subject of the formula x2y = 120.
[2]
7.
(i)
The actual distance between Cairo and Khartoum is 1580 km.
On a different map this distance is represented by 31.6 cm.
Calculate, in the form 1 : n, the scale of this map.
[2]
(ii)
A plane flies the 1580 km from Cairo to Khartoum.
It departs from Cairo at 11 55 and arrives in Khartoum at 14 03.
Calculate the average speed of the plane, in kilometres per hour.
[4]
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8.
Vreni took part in a charity walk.
She walked a distance of 20 kilometres.
On a map, the distance of 20 kilometres was represented by a length of 80 centimetres.
The scale of the map was 1 : n.
Calculate the value of n.
[2]
9.
The scale of a map is 1:20 000 000.
On the map, the distance between Cairo and Addis Ababa is 12 cm.
(i)
Calculate the distance, in kilometres, between Cairo and Addis Ababa.
[2]
(ii)
On the map the area of a desert region is 13 square centimetres.
Calculate the actual area of this desert region, in square kilometres.
[2]
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Functions
10.
f: x  5 – 3x.
(a)
Find f(–1).
Answer (a) …….………………………….
[1]
(b)
Find f –1(x).
Answer (b) …….………………………….
[2]
(c)
Find f f –1(8).
Answer (c) …….………………………….
[1]
11.
f(x) = x3 – 3x2 + 6x – 4 and g(x) = 2x – 1.
Find
(a)
f(–1),
Answer (a) ……….….…………
[1]
(b)
gf(x),
Answer (b) ……….….…………
[2]
(c)
g–1(x).
Answer (c) ……….….…………
[2]
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12.
The function f(x) is given by
f(x) = 3x – 1.
Find, in its simplest form,
(a)
f–1f(x),
Answer (a) ……………………..
[1]
(b)
ff(x).
Answer (b) ……………………..
[2]
13.
f(x) = x2 – 4x + 3
(a)
and
g(x) = 2x – 1.
Solve f(x) = 0.
[2]
(b)
Find g–1(x).
[2]
(b)
Solve f(x) = g(x), giving your answers correct to 2 decimal places.
[5]
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(d)
Find the value of gf(–2).
[2]
(e)
Find fg(x). Simplify your answer.
[3]
14.
g(x) = x2 + 1
f(x) = 2x – 1
(a)
h(x) = 2x
Find the value of
(i)
 1
f   ,
 2
Answer (a)(i) ……..……………
[1]
(ii)
g(–5),
Answer (a)(ii) ………….………
[1]
(iii)
h(–3).
Answer (a)(iii) …………………
[1]
(b)
Find the inverse function
f–1(x).
Answer (b) f–1(x) = …………..…
[2]
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(c)
g(x) = z.
Find x in terms of z.
Answer (c) x = …………....……
[2]
(d)
Find gf(x), in its simplest form.
Answer (d) gf(x) = ……….….…
[2]
(e)
h(x) = 512.
Find the value of x.
Answer (e) x = ……….…………
[1]
(f)
Solve the equation 2f(x) + g(x) = 0, giving your answers correct to 2 decimal places.
Answer (f) x = ….… or x = ….…
[5]
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(g)
Sketch the graph of
(i)
y = f(x),
(ii)
y = g(x).
y
y
x
O
O
(i) y = f(x)
x
(ii) y = g(x)
[3]
15.
f(x) = 2x – 1,
(a)
g(x) =
3
+ 1,
x
h(x) = 2x.
Find the value of fg(6).
[1]
(b)
Write, as a single fraction, gf(x) in terms of x.
[3]
(c)
Find g–1(x).
[3]
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(d)
Find hh(3).
[2]
(e)
 24 
Find x when h(x) = g   
 7 
[2]
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