Contoh Soal 4 Fungsi Komposisi dan Fungsi Invers

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

$\begin{array}{ll}\\ 17.&\textrm{Jika}\: \: f^{-1}(x)=\displaystyle \frac{x-1}{5}\: \: \textrm{dan}\: \: g^{-1}(x)=\displaystyle \frac{3-x}{2},\\ & \textrm{maka}\: \: \left (f\circ g \right )^{-1}(6)=....\\ &\begin{array}{llllll}\\ \textrm{a}.&-1&&&\textrm{d}.&2\\ \textrm{b}.&0\qquad&\color{red}\textrm{c}.&1\qquad&\textrm{e}.&3 \end{array}\\ & (\textbf{UMPTN 1995})\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{Diketahui b}&\textrm{ahwa}:\\ &\begin{cases} f^{-1}(x)&=\displaystyle \frac{x-1}{5} \\ g^{-1}(x)&=\displaystyle \frac{3-x}{2} \end{cases}\\ \left (f\circ g \right )^{-1}(x)&=\left (g^{-1}\circ f^{-1} \right )(x)\\ &=\displaystyle \frac{3-\left ( \displaystyle \frac{x-1}{5} \right )}{2}\\ \left (f\circ g \right )^{-1}(6)&=\displaystyle \frac{3-\left ( \displaystyle \frac{6-1}{5} \right )}{2}\\ &=\displaystyle \frac{3-1}{2}=\frac{2}{2}\\ &=1 \end{aligned} \end{array}$.

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