Cambridge International Mathematics

392 Introduction to functions (Chapter 19)

We write g(f(2)) = 7:

Similarly, f(g(2)) is obtained by applyinggfirst, then applyingfto the result:

2

g(x)=x+3

5

f(x)=x^2

25

3

g(x)=x+3

6

f(x)=x^2

36

We write f(g(2)) = 25 and f(g(3)) = 36:

More generally, we can define acomposite function f(g(x)) which allows us to perform calculations like

these in one step.

Given two functions f(x) and g(x), thecomposite functionoffandg

is the function which mapsxonto f(g(x)):

In the above example where f(x)=x^2 and g(x)=x+3, f(g(x)) =f(x+3)

=(x+3)^2

Notice that f(g(2)) = 25 and f(g(3)) = 36 are both true for this function.

Also notice that g(f(x)) =g(x^2 )

=x^2 +3, so in general f(g(x)) 6 =g(f(x)):

Example 6 Self Tutor

Consider the functions f(x)=x^2 +1and g(x)=2x¡ 3.

a Find the value of f(g(2)) and g(f(3)):

b Find, in simplest form, f(g(x)) and g(f(x)):

c Usebto check your answers toa.

a g(x)=2x¡ 3

) g(2) = 2(2)¡3=1

) f(g(2)) =f(1)

=1^2 +1

=2

f(x)=x^2 +1

) f(3) = 3^2 +1=10

) g(f(3)) =g(10)

= 2(10)¡ 3

=17

b f(g(x)) =f(2x¡3)

=(2x¡3)^2 +1

=4x^2 ¡ 12 x+9+1

=4x^2 ¡ 12 x+10

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

=2(x^2 +1)¡ 3

=2x^2 +2¡ 3

=2x^2 ¡ 1

c When x=2, f(g(2)) = 4(2)^2 ¡12(2) + 10

=16¡24 + 10

=2 which checks witha X

When x=3, g(f(3)) = 2(3)^2 ¡ 1

=18¡ 1

=17 which checks witha X

To find we

look at the function,

and whenever we see

we replace it by

within brackets.

fgx

f

x

gx

(())

()

IGCSE01

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Y:\HAESE\IGCSE01\IG01_19\392IGCSE01_19.CDR Monday, 13 October 2008 10:46:05 AM PETER