Fungsi Komposisi dan Fungsi Invers

$\color{blue}\textrm{A. Fungsi Komposisi}$

Perhatikanlah ilustrasi gambar berikut

$\begin{array}{|c|c|}\hline \textrm{Syarat}&\textrm{Sifat-sifat}\\\hline \begin{aligned}&R_{_{f}}\cap D_{_{g}}\neq \left \{ \: \right \} \end{aligned}&\begin{aligned}1.\: \: &\textrm{Tidak komutatif}\\ &(f\circ g)(x)\neq (g\circ f)(x)\\ 2.\: \: &\textrm{Bersifat asosiatif}\\ & f\circ (g\circ h)(x)= (f\circ g)\circ h(x)\\ 3.\: \: &\textrm{Adanya unsur dentitas}\\ & (f\circ I)(x)=(I\circ f)(x)=f(x) \end{aligned}\\\hline \end{array}$.

$\color{blue}\textrm{B. Fungsi Invers}$

$\begin{aligned}&\bullet \quad \textrm{Suatu fungsi}\: \: f:A\rightarrow B\\ &\qquad \textrm{memiliki fungsi invers} \: \: g:B\rightarrow A\\ &\qquad \textbf{jika dan hanya jika}\: \: f\\ &\qquad \textrm{merupakan fungsi}\: \textbf{bijektif}\\ &\bullet \quad \textrm{Jika fungsi}\: \: g\: \: \textrm{ada, maka}\\ &\qquad g\: \: \textrm{dinyatakan dengan}\: \: f^{-1}\: \: (\textrm{dibaca}:\: \: f\: \: \textbf{invers})\end{aligned}$.

$\begin{array}{ll}\\ &\color{blue}\textbf{Catatan}\\ &\textrm{Perlu diingat bahwa pada invers }\\ &\textrm{fungsi komposisi berlaku }\\ &\textrm{ketentuan sebagai berikut}\\ \blacklozenge &\left ( g\circ f \right )^{-1}(x)=\left ( f^{-1}\circ g^{-1} \right )(x)\\ \blacklozenge &\left ( f\circ g \right )^{-1}(x)=\left ( g^{-1}\circ f^{-1} \right )(x)\\ \blacklozenge &f(x)=\left ( \left (f^{-1} \right )^{-1}(x) \right )\\ \blacklozenge &x=f^{-1}\left ( f(x) \right )=\left ( f^{-1}\circ f \right )(x)\\ &\quad=\left ( f\circ f^{-1} \right )(x)=f\left ( f^{-1}(x) \right ) \end{array}$.

$\LARGE\colorbox{yellow}{CONTOH SOAL}$

$\begin{array}{ll}\\ 1.&\textrm{Tentukanlah}\: \: (f\circ g)(x)\: \: \textrm{dan}\: \: (g\circ f)(x)\: \: \textrm{Jika}:\\ &\textrm{a}.\quad f(x)=2-x\: \: \textrm{dah}\: \: g(x)=5x+3\\ &\textrm{b}.\quad f(x)=2x+1\: \: \textrm{dah}\: \: g(x)=x^{2}-4\\ &\textrm{c}.\quad f(x)=\displaystyle \frac{5}{x-4}\: \: \textrm{dah}\: \: g(x)=3x^{2}\\ &\textrm{d}.\quad f(x)=\sqrt{4-x}\: \: \textrm{dah}\: \: g(x)=x^{2}+x\\ &\textrm{e}.\quad f(x)=x^{3}+1\: \: \textrm{dah}\: \: g(x)=\displaystyle \frac{x}{x-1}\\ &\textrm{f}.\quad f(x)=\displaystyle \frac{3}{x-2}\: \: \textrm{dah}\: \: g(x)=\sqrt{x-4}\\ \end{array}$.

Jawab: 

hanya no. 1 a saja yang dibahas

$\begin{aligned}1.\quad \textrm{a}.\: \: \: \: (f\circ g)(x)&=f\left ( g(x) \right )\\ &=f\left ( 5x+3 \right )\\ &=2-\left ( 5x+3 \right )\\ &=-5x-1\ \textrm{dan}\: \: \: \: \: \: \: \: \: \: &\\ (g\circ f)(x)&=g\left ( f(x) \right )\\ &=g(2-x)\\ &=5(2-x)+3\\ &=10-5x+3\\ &=13-10x \end{aligned}$.

$\begin{array}{ll}\\ 2.&\textrm{Diketahui bahwa}\: \: g(x)=3x+2\\ & \textrm{dan}\: \: (g\circ f)(x)=4x-5.\: \: \textrm{Tentukanlah}\: \: f(x)\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}(g\circ f)(x)&=4x-5\\ g(f(x))&=4x-5\\ 3.f(x)+2&=4x-5\\ 3.f(x)&=4x-7\\ f(x)&=\displaystyle \frac{4x-7}{3} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 3.&\textrm{Diketahui bahwa}\: \: g(x)=x+4\: \: \textrm{dan}\\ & (f\circ g)(x)=2x^{2}+3.\: \: \textrm{Tentukanlah}\: \: f(x)\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}(f\circ g)(x)&=2x^{2}+3\\ f(g(x))&=2x^{2}+3\\ f(x+4)&=2x^{2}+3,\\ & \textrm{misalkan}\: \: x+4=a\Rightarrow x=a-4,\\ \textrm{sehingga}&,\\ f(a)&=2(a-4)^{2}+3\\ f(a)&=2(a^{2}-8a+16)+3\\ &=2a^{2}-16a+35\\ f(x)&=2x^{2}-16x+35 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 4.&\textrm{Diketahui}\: \: f(x)=3x\: \: \textrm{dan}\: \: g(x)=3^{x}.\\ &\textrm{Tentukanlah rumus untuk}\: \: ^{^{27}}\log (g\circ f)(x)\\\\ &\color{red}\textrm{Jawab}:\\ &\begin{aligned}^{^{27}}\log (g\circ f)(x)&=\: ^{^{27}}\log g(f(x))\\ &=\: ^{^{27}}\log 3^{3x}\\ &=\: ^{^{3^{3}}}\log 3^{3x}\\ &=\: ^{^{^{\left (3^{3} \right )}}}\log \left ( 3^{3} \right )^{x}\\ &=x \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 5.&\textrm{Tentukanlah invers dari}\: \: f(x)=6^{2x}\\\\ &\color{red}\textrm{Jawab}:\\ &\begin{aligned}f(x)&=6^{2x}\\ y&=6^{2x}\\ \log y&=\log 6^{2x}\\ \log y&=2x\log 6\\ \displaystyle \frac{\log y}{\log 6}&=2x\\ \displaystyle \frac{\log y}{2\log 6}&=x\\ \displaystyle \frac{\log y}{\log 6^{2}}&=x\\ \displaystyle \frac{\log y}{\log 36}&=x\\ x&=\displaystyle \frac{\log y}{\log 36}&\\ x&=\: ^{^{36}}\log y\\ f^{-1}(x)&=\: ^{^{36}}\log x \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 6.&\textrm{Tentukanlah inver dari}\\ & f(x)=\displaystyle \frac{2x+3}{4x-5},\: \: \: x\neq \displaystyle \frac{5}{4}\\\\ &\color{red}\textrm{Jawab}:\\ &\begin{aligned}f(x)&=\displaystyle \frac{2x+3}{4x-5}\\ y&=\displaystyle \frac{2x+3}{4x-5}\\ (4x-5)y&=2x+3\\ 4xy-5y&=2x+3\\ 4xy-2x&=5y+3\\ x(4y-2)&=5y+3\\ x&=\displaystyle \frac{5y+3}{4y-2}\\ f^{-1}(y)&=\displaystyle \frac{5y+3}{4y-2}\\ \textrm{maka},&\\ f^{-1}(x)&=\displaystyle \frac{5x+3}{4x-2},\: \: \: x\neq \displaystyle \frac{1}{2} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 7.&\textrm{Jika}\: \: f(x)=-2x-4\: \: \textrm{dan}\: \: g(x)=\displaystyle \frac{20-3x}{2},\\ & \textrm{maka nilai dari}\: \: (f\circ g)^{-1}(2)=....\\\\ &\color{blue}\textrm{Jawab}:\\ &\textbf{Perhatikan bahwa}\\ &\begin{aligned}&\\ &(f\circ g)^{-1}(x)=\left ( g^{-1}\circ f^{-1} \right )(x)\\ & \end{aligned}\\ &\begin{array}{|c|c|}\hline (f\circ g)^{-1}(x)&\left ( g^{-1}\circ f^{-1} \right )(x)\\\hline \begin{aligned}(f\circ g)(x)&=f(g(x))\\ y&=-2\left ( \displaystyle \frac{20-3x}{2} \right )-4\\ y&=3x-20-4\\ y&=3x-24\\ y+24&=3x\\ x&=\displaystyle \frac{y+24}{3}\\ (f\circ g)^{-1}y&=\displaystyle \frac{y+24}{3}\\ (f\circ g)^{-1}(2)&=\displaystyle \frac{2+24}{3}\\ &=\displaystyle \frac{26}{3} \end{aligned}&\begin{aligned}\left ( g^{-1}\circ f^{-1} \right )(x)&=g^{-1}\left ( f^{-1}(x) \right )\\ &=......\\ &=....\\ &=....\\ &=....\\ &=....\\ &=....\\ &=....\\ &=....\\ &=....\\ \left ( g^{-1}\circ f^{-1} \right )(2)&=....\\ &=.... \end{aligned}\\\hline \end{array} \end{array}$.

DAFTAR PUSTAKA

1. Soedyarto, Nugroho, Maryanto. 2008. Matematika 2 untuk SMA dan MA Kelas XI Program IPA. Jakarta: Pusat Perbukuan Departemen Pendidikan Nasional.
2. Sunardi, Waluyo, S., Sutrisno, & Subagya. 2005. Matematika 2untuk SMA Kelas 2 IPA. Jakarta: BUMI AKSARA.