Cambridge International Mathematics

10 Consider =

x+2

x¡ 1

and g(x)=2x.

a Sketch the graphs of y=f(x) and y=g(x) on the same set of axes for ¡ 56 x 65

b Find the coordinates of the points of intersection of the two graphs.

c Find the values ofxfor which 2 x>

x+2

x¡ 1

:

SOLVING UNFAMILIAR EQUATIONS

Technology allows us to solve equations with expressions we are unfamiliar with.

Suppose we are given an equation of the form f(x)=g(x):

If we subtract g(x) from both sides, we have f(x)¡g(x)=0.

So, given f(x)=g(x), there are two different approaches we can take to find the solutions.

Method 1: Graph y=f(x) and y=g(x) on the same set of axes and find thex-coordinates

where they meet.

Method 2: Graph y=f(x)¡g(x) and find thex-intercepts.

As an example we will consider an equation that we can solve algebraically:

2 x^2 =3x+2

) 2 x^2 ¡ 3 x¡2=0

) (2x+ 1)(x¡2) = 0

) x=¡^12 or 2

Consider f(x)=2x^2 and g(x)=3x+2.

Method 1:

We graphy=f(x) andy=g(x) on the same

set of axes.

The graphs intersect at the points with

x-coordinates¡^12 and 2.

Method 2:

We graph y=f(x)¡g(x) which is

y=2x^2 ¡ 3 x¡ 2

Thex-intercepts are¡^12 and 2.

y

x

O

&-\Qw_\'\\Qw_\*

()2 ¡8,

yx¡=¡2X

yx¡=¡3 ¡+¡2

y

x

-\Qw_2O

yx x¡=¡2 ¡-3 ¡-¡2X

.

f(x)

478 Further functions (Chapter 23)

IGCSE01

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Y:\HAESE\IGCSE01\IG01_23\478IGCSE01_23.CDR Monday, 27 October 2008 2:18:51 PM PETER