Cambridge International Mathematics

Introduction to functions (Chapter 19) 393

EXERCISE 19D

1 If f(x)=x^2 +1and g(x)=2x¡ 3 find:

a f(3) b g(3) c f(g(3)) d g(f(3)) e f(f(3)) f g(g(3))

2 Let f(x)=5x+1and g(x)=4¡ 2 x.

a Find the values of f(g(0)) and g(f(1)):

b Find in simplest form: i f(g(x)) ii g(f(x))

c Useato check your answers tob.

3 Let f(x)=x^2 +2xand g(x)=x¡ 2. Find, in simplest form:

a f(g(2)) and f(g(x)) b g(f(2)) and g(f(x)).

4 Iff(x)=3x¡ 4 and g(x)=2¡x, find in simplest form:

a f(g(x)) b g(f(x)) c f(f(x)) d g(g(x))

5 Let f(x)=x^2 +x¡ 6 and g(x)=x+2.

a Find f(g(x)) in simplest form.

b Useato find:

i f(g(1)) ii f(g(4)) iii f(g(¡2)):

6 Iff(x)=

p

x andg(x)=4x¡ 3 , find in simplest form:

a f(g(x)) b g(g(x))

7 Find anfand agfunction such that:

a f(g(x)) =

p

x¡ 3 b f(g(x)) = (x+5)^3 c f(g(x)) =

5

x+7

d g(f(x)) =

1

p

3 ¡ 4 x

efg(f(x)) =

4

(x¡1)^2

8

9

10

11

Areciprocal functionhas an equation of the form y=

k

x

wherekis a constant.

It has a graph which is called arectangular hyperbola.

There are examples of hyperbolae in the world around us.

When an aeroplane flies faster than the speed of sound, which is around 1200 km/h, we say it breaks the

sound barrier. It sets up a shock wave in the shape of acone, and when this intersects the ground it does

so in the shape of ahyperbola. Thesonic boomformed hits every point on the curve at the same time, so

that people in different places along the curve on the ground all hear it at the same time. No sound is heard

outside the curve but the boom eventually covers every place inside it.

E RECIPROCAL FUNCTIONS [3.2, 3.5]

g(f(x)) = 3^2 x+1

Iff(x)=2x+1and g(f(x)) = 4x^2 +4x+3, find g(x), given that g(x)=ax^2 +bx+c.

Ifg(x)=1¡ 3 x andf(g(x)) = 9x^2 ¡ 6 x¡ 2 , find f(x), given that f(x)=ax^2 +bx+c.

Iff(x)=3x+1and g(x)=2x¡ 3 , findxwhen g(f(x)) = 17:

If f(x)=ax+b and f(f(x)) = 4x¡ 15 , findaandb.

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Y:\HAESE\IGCSE01\IG01_19\393IGCSE01_19.CDR Tuesday, 21 October 2008 2:10:53 PM PETER