1.6.4 Function, SPM Practice (Short Question)

Question 10 (4 marks):
Given the function g : x → 2x – 8, find
$\begin{array}{l}\left(\text{a}\right)g^{-1}\left(x\right),\\ \left(\text{b}\right)\text{ the value of }p\text{ such that }{g}^2\left(\frac{3p}{2}\right)=30.\end{array}$

Solution:
(a)
$\begin{array}{l}\text{Let }y=g\left(x\right)\\ =2x-8\\ 2x-8=y\\ 2x=y+8\\ x=\frac{y+8}{2}\\ \text{Thus, }{g}^{-1}\left(x\right)=\frac{x+8}{2}\end{array}$

(b)
$\begin{array}{l}g\left(x\right)=2x-8\\ g^2\left(x\right)=g\left[g\left(x\right)\right]\\ =g\left(2x-8\right)\\ =2\left(2x-8\right)-8\\ =4x-16-8\\ =4x-24\\ g^2\left(\frac{3p}{2}\right)=30\\ 4\left(\frac{3p}{2}\right)-24=30\\ 6p=54\\ p=9\end{array}$

Question 11 (4 marks):
Diagram 9 shows the relation between set A, set B and set C.

Diagram 9

It is given that set A maps to set B by the function $\frac{x+1}{2}$ and maps to set C by fg : x → x² + 2x + 4.
(a) Write the function which maps set A to set B by using the function notation.
(b) Find the function which maps set B to set C.


Solution:

(a)
$g:x\to \frac{x+1}{2}$

(b)

$\begin{array}{l}g\left(x\right)=\frac{x+1}{2}\\ fg\left(x\right)=x^2+2x+4\\ f\left[g\left(x\right)\right]=x^2+2x+4\\ f\left(\frac{x+1}{2}\right)=x^2+2x+4\\ \\ \text{Let }\frac{x+1}{2}=y\\ x+1=2y\\ x=2y-1\\ \\ \therefore f\left(y\right)={\left(2y-1\right)}^2+2\left(2y-1\right)+4\\ f\left(y\right)=4y^2-4y+1+4y-2+4\\ f\left(y\right)=4y^2+3\\ f\left(x\right)=4x^2+3\\ \\ \text{Thus, function which maps set }B\text{ to set }C\\ \text{is }f\left(x\right)=4x^2+3\end{array}$