Analysis and Approaches for IBDP Maths Ebook 2 | Page 75

24
Paper 2 Section A – Symmetric Properties
Example
The function f and g are defined by
f ( x) asin
x bx
2
� � and
a and b are constants. It is given that f is an odd function.
g( x) 4
3
� x , where x� ,
(a) Find the value of a and the value of b .
Let h( x) � f ( x) � g( x)
.
[3]
(b)
Solution
Show that h is an even function.
[2]
7
(a) f ( �x) � � f ( x)
a �x �b �x � � a x � bx
(M1) for valid approach
2 2
sin( ) ( ) ( sin )
2 2
�asin
x �bx � �asin
x � bx
a� A1
bx
�� bx
2 2
2
2bx � 0
b � 0
A1
(b) h( �x) � f ( �x) � g( � x)
M1
h x f x x
3
( � ) � � ( ) �(4( � ) )
h �x � � f x � � x
A1
3
( ) ( ) ( 4 )
h( �x) � f ( x) � 4x
h( �x) � f ( x) � g( x)
h( �x) � h( x)
Thus, h is an even function.
3
AG
[3]
[2]
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