396 Introduction to functions (Chapter 19)
2 Draw the graph of y=
(
x if x> 0
¡x if x< 0
using the results of 1.
3 Consider M(x)=
p
x^2 wherex^2 must be found before finding the square root.
Find the values of M(5),M(7),M(^12 ),M(0),M(¡ 2 ),M(¡8)andM(¡10).
4 What conclusions can be made from 1 and 3?
THE ABSOLUTE VALUE OF A NUMBER
Theabsolute valueormodulusof a real number is its size, ignoring its sign.
We denote the absolute value ofxbyjxj.
For example, the absolute value of 7 is 7 , and
the absolute value of¡ 7 is also 7.
Geometric definition of absolute value
If x> 0 :Ifx< 0 :
For example:
Algebraic definition of absolute value
FromDiscovery 3,
The vertical line
x=0is the
line of symmetry
of the graph.
Example 7 Self Tutor
a ja+bj
=j¡7+3j
=j¡ 4 j
=4
b jabj
=j¡ 7 £ 3 j
=j¡ 21 j
=21
The absolute value
behaves as a grouping
symbol. Perform all
operations within it first.
77
-7 07
||x
0 x
||x
x 0
y
x
This branch
isyxx= -, <0.
This branch
isyxx= , >0.
O
jxjis the distance ofxfrom 0 on the number line.
Because the modulus is a distance, it cannot be negative.
jxj=
(
x ifx> 0
¡x ifx< 0
or jxj=
p
x^2
y=jxj has graph:
If a=¡ 7 and b=3find: a ja+bj b jabj
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y:\HAESE\IGCSE01\IG01_19\396IGCSE01_19.cdr Friday, 10 October 2008 10:17:52 AM PETER