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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+2Let g1(x)=yg(y)=x3y+2=xy=x23g1(x)=x23

(a)(ii)
hg(x)=12x+5h(3x+2)=12x+5g(x)=3x+2Let u=3x+2   x=u23h(u)=12(u23)+5   =4u8+5   =4u3h(x)=4x3

(b)
gh(x)=g(4x3) =3(4x3)+2 =12x9+2 =12x712x7=8x+1   4x=8 x=2


Question 6:
Given that f:x5xmx3,x32 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=0m(32)3=0m=3×23=2

(b)
Let y=5x2x32xy3y=5x2xy5x=3yx(2y5)=3y x=3y2y5f1(y)=3y2y5f1(x)=3x2x5f1(2)=3(2)2(2)5=23

gf1(2)=g(23)=34(23)=13

(c)
hg (x) = 12x + 5
h [g(x)] = 12x + 5
(3 – 4x) = 12x + 5
Let u = 3 – 4x
x=3u4h(u)=12(3u4)+5h(u)=93u+5h(u)=143uh(x)=143x

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