**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*) = 8

*x*+ 1.

*Solution*:**(a)(i)**

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

**(a)(ii)**

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

**(b)**

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

(a)(1)

g-1(x)= x-2 over 3

Thanks for pointing out our mistake, correction had been made accordingly.