Cambridge International AS and A Level Mathematics Pure Mathematics 1

Functions

P1^

4

Composite functions

It is possible to combine functions in several different ways, and you have already

met some of these. For example, if f(x) = x^2 and g(x) = 2 x, then you could write

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

In this example, two functions are added.

Similarly if f(x) = x and g(x) = sin x, then

f(x).g(x) = x sin x.

In this example, two functions are multiplied.

Sometimes you need to apply one function and then apply another to the answer.

You are then creating a composite function or a function of a function.

ExaMPlE 4.2 A new mother is bathing her baby for the first time. She takes the temperature

of the bath water with a thermometer which reads in Celsius, but then has to

convert the temperature to degrees Fahrenheit to apply the rule that her own

mother taught her:

At one o five

He’ll cook alive

But ninety four

is rather raw.

Write down the two functions that are involved, and apply them to readings of

(i) 30°C (ii) 38°C (iii) 45°C.

SOlUTION

The first function converts the Celsius temperature C into a Fahrenheit

temperature, F.

F =

9

5

C

+ 32

The second function maps Fahrenheit temperatures on to the state of the bath.

F  94 too cold

94  F  105 all right

F  105 too hot

This gives

(i) 30°C  86°F  too cold

(ii) 38°C  100.4°F  all right

(iii) 45°C  113°C  too hot.