1.6.2 Fungsi, SPM Praktis (Soalan Panjang)

Soalan 3:
Fungsi f dan g ditakrifkan oleh
$\begin{array}{l}f:x\mapsto 2x-3\\ g:x\mapsto \frac{2}{x};x\ne 0\end{array}$
Ungkapkan dalam bentuk yang serupa
(a) ff,
(b) gf,
(c) f² , Hitungkan nilai x supaya ff(x) = gf(x).

Penyelesaian:
(a)
$\begin{array}{l}ff\left(x\right)=f\left[f\left(x\right)\right]\\ =f\left(2x-3\right)\\ =2\left(2x-3\right)-3\\ =4x-9\\ \text{Jadi, }ff:x\mapsto 4x-9\end{array}$


(b)
$\begin{array}{l}gf\left(x\right)=g\left[f\left(x\right)\right]\\ =g\left(2x-3\right)\\ =\frac{2}{2x-3}\\ \text{Jadi, }gf:x\mapsto \frac{2}{2x-3}\end{array}$

(c)
$\begin{array}{l}\text{Katakan }{f}^{-1}\left(x\right)=y,\\ \text{maka}f\left(y\right)=x\\ 2y-3=x\\ y=\frac{x+3}{2}\\ \text{maka }{f}^{-1}\left(x\right)=\frac{x+3}{2}\\ f^{-1}:x\mapsto \frac{x+3}{2}\\ \\ \text{Apabila }ff\left(x\right)=gf\left(x\right),\\ 4x-9=\frac{2}{2x-3}\\ \left(4x-9\right)\left(2x-3\right)=2\\ 8x^2-30x+27=2\\ 8x^2-30x+25=0\\ \left(4x-5\right)\left(2x-5\right)=0\\ 4x-5=0\text{ atau }2x-5=0\\ x=\frac{5}{4}\text{ atau }x=\frac{5}{2}\end{array}$

Soalan 4:
Fungsi f dan g ditakrifkan oleh
$\begin{array}{l}f\left(x\right)=3x-2\\ g\left(x\right)=\frac{3}{x},x\ne 0\\ \text{Cari}\\ \text{(a) }{f}^{-1}\left(2\right),\\ \text{(b) g}f\left(-3\right),\\ \text{(c) fungsi }h\text{ jika diberi }hf\left(x\right)=3x+2,\\ \text{(d) fungsi }k\text{ jika diberi }fk\left(x\right)=4x-7.\end{array}$

Penyelesaian:
(a)
$\begin{array}{l}\text{Katakan }{f}^{-1}\left(2\right)=x,\\ \text{maka}f\left(x\right)=2\\ \text{ 3}x-2=2\\ 3x=4\\ x=\frac{4}{3}\\ f^{-1}\left(2\right)=\frac{4}{3}\end{array}$

(b)
$\begin{array}{l}gf\left(-3\right)=g\left[3\left(-3\right)-2\right]\\ =g\left(-11\right)\\ =-\frac{3}{11}\end{array}$

(c)
$\begin{array}{l}h\left[f\left(x\right)\right]=3x+2\\ h\left(3x-2\right)=3x+2\\ \text{Katakan }y=3x-2\\ \text{Maka }x=\frac{y+2}{3}\\ h\left(y\right)=3\left(\frac{y+2}{3}\right)+2\\ =y+2+2\\ =y+4\\ \text{Maka }h\left(x\right)=x+4\end{array}$

(d)
$\begin{array}{l}f\left[k\left(x\right)\right]=4x-7\\ 3k\left(x\right)-2=4x-7\\ 3k\left(x\right)=4x-5\\ k\left(x\right)=\frac{4x-5}{3}\end{array}$