1.7.3 Function, SPM Practice (Long Question)


Question 5:
In diagram below, the function g maps set P to set Q and the function h maps set Q to set R.



Find
(a) in terms of x, the function
(i) which maps set Q to set P,
(ii) h(x).
(b) the value of x such that gh(x) = 8x + 1.



Solution:
(a)(i)
g( x )=3x+2 Let  g 1 ( x )=y g( y )=x 3y+2=x y= x2 3 g 1 ( x )= x2 3

(a)(ii)
hg( x )=12x+5 h( 3x+2 )=12x+5 g( x )=3x+2 Let u=3x+2    x= u2 3 h( u )=12( u2 3 )+5    =4u8+5    =4u3 h( x )=4x3

(b)
gh( x )=g( 4x3 )  =3( 4x3 )+2  =12x9+2  =12x7 12x7=8x+1    4x=8  x=2


Question 6:
Given that f:x 5x mx3 ,x 3 2  and g:x34x. Find
(a) the value of m, [2 marks]
(b)   gf-1(–2), [3 marks]
(c) function h if hg (x) = 12x + 5 [3 marks]

Solution:
(a)
mx3=0 m( 3 2 )3=0 m=3× 2 3 =2

(b)
Let y= 5x 2x3 2xy3y=5x 2xy5x=3y x( 2y5 )=3y  x= 3y 2y5 f 1 ( y )= 3y 2y5 f 1 ( x )= 3x 2x5 f 1 ( 2 )= 3( 2 ) 2( 2 )5 = 2 3

g f 1 ( 2 )=g( 2 3 ) =34( 2 3 ) = 1 3

(c)
hg (x) = 12x + 5
h [g(x)] = 12x + 5
(3 – 4x) = 12x + 5
Let u = 3 – 4x
x= 3u 4 h( u )=12( 3u 4 )+5 h( u )=93u+5 h( u )=143u h( x )=143x

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