1.6.3 Fungsi, SPM Praktis (Soalan Kertas 2)

Soalan 5:
$\begin{array}{l}\text{Fungsi }f\text{ ditakrifkan oleh }f:x\to \frac{1+x}{1-x},x\ne 1.\\ \text{Cari }{f}^2,f^3,f^4\text{ dan seterusnya tulis fungsi bagi }{f}^{51}\text{ dan }{f}^{52}\text{.}\end{array}$

Penyelesaian:
$\begin{array}{l}f\left(x\right)=\frac{1+x}{1-x},x\ne 1\\ f^2\left(x\right)=f\left[f\left(x\right)\right]=f\left(\frac{1+x}{1-x}\right)\\ =\frac{1+\left(\frac{1+x}{1-x}\right)}{1-\left(\frac{1+x}{1-x}\right)}=\frac{\frac{1-x+1+x}{\cancel{1-x}}}{\frac{1-x-1-x}{\cancel{1-x}}}\\ =\frac{2}{-2x}=-\frac{1}{x}\\ \\ f^3\left(x\right)=f\left[f^2\left(x\right)\right]=f\left(-\frac{1}{x}\right)\\ =\frac{1+\left(-\frac{1}{x}\right)}{1-\left(-\frac{1}{x}\right)}=\frac{\frac{x-1}{\cancel{x}}}{\frac{x+1}{\cancel{x}}}\\ =\frac{x-1}{x+1}\\ \\ f^4\left(x\right)=f\left[f^3\left(x\right)\right]=f\left(\frac{x-1}{x+1}\right)\\ =\frac{1+\left(\frac{x-1}{x+1}\right)}{1-\left(\frac{x-1}{x+1}\right)}=\frac{\frac{x+1+x-1}{\cancel{x+1}}}{\frac{x+1-x+1}{\cancel{x+1}}}\\ =\frac{2x}{2}=x\\ \\ f^5\left(x\right)=f\left[f^4\left(x\right)\right]=f\left(x\right)=\frac{1+x}{1-x}\left(\text{berulang}\right)\\ \\ \therefore f^{51}\left(x\right)=f^3\left[f^{48}\left(x\right)\right]=f^3\left(x\right)\\ =\frac{x-1}{x+1}\\ f^{52}\left(x\right)=f^4\left[f^{48}\left(x\right)\right]=f^4\left(x\right)=x\end{array}$

Soalan 6:
Dalam rajah di bawah, fungsi g memetakan set P kepada set Q dan fungsi h memetakan set Q kepada set R.



Cari
(a) dalam sebutan x, fungsi
(i) yang memetakan set Q kepada set P,
(ii) h(x).

(b) nilai x dengan keadaan gh(x) = 8x + 1.


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

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

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