Cambridge Additional Mathematics

Functions (Chapter 2) 37

RELATIONS

InChapter 1, we saw that:

Arelationis any set of points which connect two variables.

A relation is often expressed in the form of anequationconnecting thevariablesxandy. The relation is

a set of points(x,y)

For example, y=x+3 and x=y^2 are the equations of two relations. Each equation generates a set

of ordered pairs, which we can graph.

y=x+3is a set

of points which lie

in a straight line

x=y^2 is a set of

points which lie in

a smooth curve.

FUNCTIONS

Afunction, sometimes called amapping, is a relation in which no two

different ordered pairs have the samex-coordinate or first component.

We can see from the above definition that a function is a special type of relation.

Every function is a relation, but not every relation is a function.

TESTING FOR FUNCTIONS

Algebraic Test:

If a relation is given as an equation, and the substitution of any value forx

results in one and only one value ofy, then the relation is a function.

For example:

² y=3x¡ 1 is a function, since for any value ofxthere is only one corresponding value ofy

² x=y^2 is not a function, since if x=4then y=§ 2.

Geometric Test or Vertical Line Test:

Suppose we draw all possible vertical lines on the graph of a relation.

² If each line cuts the graph at most once, then the relation is a function.

² If at least one line cuts the graph more than once, then the relation isnota function.

GRAPHICAL NOTE

² If a graph contains a smallopen circlesuch as , this point isnot included.

² If a graph contains a smallfilled-in circlesuch as , this pointis included.

² If a graph contains anarrow headat an end such as , then the graph continues indefinitely in

that general direction, or the shape may repeat as it has done previously.

y

x

3

-3

y=x+3

O

y

x

x=y¡¡ 2

2

4

-2

O

which can be viewed in theCartesian plane.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_02\037CamAdd_02.cdr Thursday, 19 December 2013 1:29:34 PM BRIAN