Cambridge International Mathematics

Introduction to functions (Chapter 19) 391

Discovery 1 Fluid filling functions

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When water is added to a container, the depth of water is given by a function of time. If

the water is added at aconstant rate, the volume of water added is directly proportional

to the time taken to add it.

So, for a container with a uniform cross-section, the graph of depth against time is a

straight line, orlinear. We can see this in the following depth-time graph.

The question arises:

‘How does the shape of the container affect the

appearance of the graph?’

For example, consider the graph shown for a vase of

conical shape.

What to do:

1 For each of the following containers, draw a depth-time graph as water is added at a constant rate.

abcd

efgh

2 Use the water filling demonstration to check your answers to question 1.

Consider a process where a value such as 2 is applied to a functionf, and then the result is applied to

another functiong:

2

f

f(2)

g

g(f(2))

The resulting value is g(f(2)):

For example, suppose 2 is applied to the functionf(x)=x^2 , and then the result is applied to g(x)=x+3:

2

f(x)=x^2

4

g(x)=x+3

7

D COMPOSITE FUNCTIONS [3.7]

DEMO

water

depth

time

depth

time

depth

O

O

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y:\HAESE\IGCSE01\IG01_19\391IGCSE01_19.CDR Friday, 10 October 2008 10:04:36 AM PETER