Contoh Soal 4 Fungsi Komposisi dan Fungsi Invers

$\begin{array}{l}\\ 16.& \text{Jika}f\text{dan}g\text{adalah fungsi yang }\\ & \text{mempunyai invers dan memenuhi}\\ & f(2x)=g(x-3),\text{maka}{f}^{-1}(x)\\ & \text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& g^{-1}\left({\displaystyle \frac{x}{2}-\frac{2}{3}}\right)\\ \text{b}.& g^{-1}\left({\displaystyle \frac{x}{2}}\right)-{\displaystyle \frac{2}{3}}\\ \text{c}.& g^{-1}(2x+6)\\ \text{d}.& 2g^{-1}(x)-6\\ \text{e}.& 2g^{-1}(x)+6\end{array}\\ & (\mathbf{\text{SBMPTN 2016 Mat Das}})\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& \text{Misalkan bahwa}\\ & f(2x)=g(x-3)=x,\\ & \text{maka}\\ & \{\begin{array}{ll}{f}^{-1}(x)& =2x\\ g^{-1}(x)& =x-3\end{array}\end{array}\\ & \begin{array}{|cc|}\hline \text{Sintak}& \text{Hasil}\\ \begin{array}{rl}{g}^{-1}(x)& =x-3\\ x& =g^{-1}(x)+3\\ & \end{array}& \begin{array}{rl}{f}^{-1}(x)& =2x\\ & =2(g^{-1}(x)+3)\\ & =2g^{-1}(x)+6\end{array}\\ \hline\end{array}\end{array}$.

$\begin{array}{l}\\ 17.& \text{Jika}{f}^{-1}(x)={\displaystyle \frac{x-1}{5}\text{dan}{g}^{-1}(x)={\displaystyle \frac{3-x}{2},}}\\ & \text{maka}{(f\circ g)}^{-1}(6)=....\\ & \begin{array}{l}\\ \text{a}.& -1& & & \text{d}.& 2\\ \text{b}.& 0& \text{c}.& 1& \text{e}.& 3\end{array}\\ & (\mathbf{\text{UMPTN 1995}})\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}\text{Diketahui b}& \text{ahwa}:\\ & \{\begin{array}{ll}{f}^{-1}(x)& ={\displaystyle \frac{x-1}{5}}\\ g^{-1}(x)& ={\displaystyle \frac{3-x}{2}}\end{array}\\ {(f\circ g)}^{-1}(x)& =(g^{-1}\circ f^{-1})(x)\\ & ={\displaystyle \frac{3-\left({\displaystyle \frac{x-1}{5}}\right)}{2}}\\ {(f\circ g)}^{-1}(6)& ={\displaystyle \frac{3-\left({\displaystyle \frac{6-1}{5}}\right)}{2}}\\ & ={\displaystyle \frac{3-1}{2}=\frac{2}{2}}\\ & =1\end{array}\end{array}$.

$\begin{array}{l}\\ 18.& \text{Invers dari}f(x)={125}^x\text{adalah}{f}^{-1}(x)\\ & \text{Nilai dari}{f}^{-1}\left(5\sqrt{5}\right)=....\\ & \begin{array}{l}\\ \text{a}.& 1& & & \text{d}.& {\displaystyle \frac{3}{5}}\\ \text{b}.& {\displaystyle \frac{1}{2}}& \text{c}.& {\displaystyle \frac{1}{6}}& \text{e}.& -{\displaystyle \frac{1}{2}}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& \text{Diketahui bahwa}f(x)={125}^x,\text{maka}\\ & f(x)=y={125}^x\\ & \text{Kedua ruas dilogkan masing-masing}\\ & \log y=\log {125}^x\\ & \iff \log y=x\log 125\\ & \iff x\log 125=\log y\\ & \iff x={\displaystyle \frac{\log y}{\log 125}}\\ & \iff x={}^{125}\log y\\ & \iff f^{-1}(x)={}^{125}\log x\\ & \text{Selanjutnya}\\ & f^{-1}(x)={}^{125}\log x\\ & \iff f^{-1}(5\sqrt{5})={}^{125}\log (5\sqrt{5})\\ & \iff f^{-1}(5\sqrt{5})={}^{5^3}\log 5^{{.}^{\frac{3}{2}}}\\ & \iff f^{-1}(5\sqrt{5})={\displaystyle \frac{{\displaystyle \frac{3}{2}}}{3}{}^5\log 5}\\ & \iff f^{-1}(5\sqrt{5})={\displaystyle \frac{1}{2}.1}\\ & \iff f^{-1}(5\sqrt{5})={\displaystyle \frac{1}{2}}\end{array}\end{array}$