Question 5:
In diagram below, the function g maps set P to set Q and the function h maps set Q to set R.
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2018/06/Picture1.png)
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 g−1(x)=yg(y)=x3y+2=xy=x−23g−1(x)=x−23
(a)(ii)
hg(x)=12x+5h(3x+2)=12x+5→g(x)=3x+2Let u=3x+2 x=u−23h(u)=12(u−23)+5 =4u−8+5 =4u−3h(x)=4x−3
(b)
gh(x)=g(4x−3) =3(4x−3)+2 =12x−9+2 =12x−712x−7=8x+1 4x=8 x=2
In diagram below, the function g maps set P to set Q and the function h maps set Q to set R.
![](http://spmaddmaths.blog.onlinetuition.com.my/wp-content/uploads/sites/3/2018/06/Picture1.png)
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 g−1(x)=yg(y)=x3y+2=xy=x−23g−1(x)=x−23
(a)(ii)
hg(x)=12x+5h(3x+2)=12x+5→g(x)=3x+2Let u=3x+2 x=u−23h(u)=12(u−23)+5 =4u−8+5 =4u−3h(x)=4x−3
(b)
gh(x)=g(4x−3) =3(4x−3)+2 =12x−9+2 =12x−712x−7=8x+1 4x=8 x=2
Question 6:
gf−1(−2)=g(23)=3−4(23)=13
Given that f:x→5xmx−3,x≠32 and g:x→3−4x. 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)
mx−3=0m(32)−3=0m=3×23=2
(b)
Let y=5x2x−32xy−3y=5x2xy−5x=3yx(2y−5)=3y x=3y2y−5f−1(y)=3y2y−5f−1(x)=3x2x−5f−1(−2)=3(−2)2(−2)−5=23
gf−1(−2)=g(23)=3−4(23)=13
(c)
hg (x) = 12x + 5
h [g(x)] = 12x + 5
h (3 – 4x) = 12x + 5
Let u = 3 – 4x
x=3−u4h(u)=12(3−u4)+5h(u)=9−3u+5h(u)=14−3uh(x)=14−3x
(a)(1)
g-1(x)= x-2 over 3
Thanks for pointing out our mistake, correction had been made accordingly.