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)
$\begin{array}{l}g\left(x\right)=3x+2\\ \text{Let }{g}^{-1}\left(x\right)=y\\ g\left(y\right)=x\\ 3y+2=x\\ y=\frac{x-2}{3}\\ g^{-1}\left(x\right)=\frac{x-2}{3}\end{array}$

(a)(ii)
$\begin{array}{l}hg\left(x\right)=12x+5\\ h\left(3x+2\right)=12x+5\to \boxed{g\left(x\right)=3x+2}\\ \text{Let }u=3x+2\\ x=\frac{u-2}{3}\\ h\left(u\right)=12\left(\frac{u-2}{3}\right)+5\\ =4u-8+5\\ =4u-3\\ h\left(x\right)=4x-3\end{array}$

(b)
$\begin{array}{l}gh\left(x\right)=g\left(4x-3\right)\\ =3\left(4x-3\right)+2\\ =12x-9+2\\ =12x-7\\ 12x-7=8x+1\\ 4x=8\\ x=2\end{array}$