Contoh Soal 1 Fungsi Komposisi dan Fungsi Invers

 $\begin{array}{ll}\\ 1.&\textrm{Diketahui fungsi}\: \: f(2x)=8x-9\\ & \textrm{dan}\: \: g(3x+1)=6x+3.\\ &\textrm{Rumus untuk}\: \: \left ( f+g \right )(x)=....\\ &\begin{array}{llllll}\\ \textrm{a}.&6x+8&&&\textrm{d}.&14x-6\\ \color{red}\textrm{b}.&6x-8&\textrm{c}.&14x+6&\textrm{e}.&6x-6 \end{array}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Diketahui}\: \textrm{bahwa}:\\ &\begin{cases} &\color{blue}f(2x) =8x-9\\ &\Rightarrow \quad f(x)=f\left ( 2\left ( \displaystyle \frac{x}{2} \right ) \right )\\ &=8\left ( \displaystyle \frac{x}{2} \right )-9=4x-9 \\ &\color{blue}g(3x+1) =6x+3\\ &\Rightarrow \quad g(x)=g\left ( 3\left ( \displaystyle \frac{x-1}{3} \right )+1 \right )\\ &=6\left ( \displaystyle \frac{x-1}{3} \right )+3\\ &=2x+1 \end{cases}\\ &\left ( f+ g \right )(x)=(4x-9)+(2x+1)\\ &=6x-8 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 2.&\textrm{Diketahui fungsi}\: \: f(x)=2x-1\\ & \textrm{dan}\: \: g(x)=x^{2}.\: \: \textrm{Fungsi}\: \: (f+g)(x^{2})=\: ....\\ &\begin{array}{llllll}\\ \textrm{a}.&x^{2}+2x-1\\ \color{red}\textrm{b}.&x^{4}+2x^{2}-1\\ \textrm{c}.&x^{4}+2x-1\\ \textrm{d}.&x^{4}+(2x-1)^{2}\\ \textrm{e}.&x^{4}+2x \end{array}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Diketahui}\: \textrm{bahwa}:\\ &\begin{cases} &\color{blue}f(x) =2x-1\\ &\color{blue}g(x) =x^{2} \end{cases}\\ &\left ( f+ g \right )(x)=(2x-1)+(x^{2})\\ &=x^{2}+2x-1\\ &\textrm{maka}\\ &(f+g)(x^{2})=(x^{2})^{2}+2(x^{2})-1\\ &\: \: \: \: \qquad\qquad =x^{4}+2x^{2}-1 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 3.&\textrm{Jika}\: \: f(x)=3-x,\: \: \textrm{maka}\\ & f\left ( x^{2} \right )+\left (f(x) \right )^{2}-2f(x)=....\\ &\begin{array}{llllll}\\ \textrm{a}.&2x^{2}-6x+4\\ \textrm{b}.&2x^{2}+4x+6\\ \textrm{c}.&2x^{2}-4x-6\\ \textrm{d}.&6x+4\\ \color{red}\textrm{e}.&-4x+6 \end{array}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Diketahui bahwa}\\ & f(x)=3-x,\: \textrm{sehingga}\\ &f\left ( x^{2} \right )+\left (f(x) \right )^{2}-2f(x)\\ &=\left ( 3-x^{2} \right )+\left ( 3-x \right )^{2}-2(3-x)\\ &=\left (3-x^{2} \right )+\left ( 9-6x+x^{2} \right )-\left ( 6-2x \right )\\ &=-x^{2}+x^{2}-6x+2x+3+9-6\\ &=-4x+6 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 4.&\textrm{Diketahui fungsi}\: \: f:\mathbb{R}\rightarrow \mathbb{R}\: \: \textrm{dan}\\ & g:\mathbb{R}\rightarrow \mathbb{R}\: \: \textrm{dirumuskan dengan}\\ &f(x)=x-1\: \: \textrm{dan}\: \: g(x)=x^{2}+2x-3.\\ & \textrm{Fungsi komposisi}\: \: g\: \: \textrm{atas}\: \: f\\ &\textrm{dinotasikan dengan}\\ &\begin{array}{ll}\\ \color{red}\textrm{a}.&\left ( g\circ f \right )(x)=x^{2}-4\\ \textrm{b}.&\left ( g\circ f \right )(x)=x^{2}-5\\ \textrm{c}.&\left ( g\circ f \right )(x)=x^{2}-6\\ \textrm{d}.&\left ( g\circ f \right )(x)=x^{2}-4x-4\\ \textrm{e}.&\left ( g\circ f \right )(x)=x^{2}-4x-5 \end{array}\\ & (\textbf{UN 2016})\\\\ &\textrm{Jawab}:\\ &\begin{aligned}\textrm{Diketahui}&\: \textrm{bahwa}:\\ &\begin{cases} \color{blue}f(x) &\color{blue}=x-1 \\ \color{blue}g(x) &\color{blue}=x^{2}+2x-3 \end{cases}\\ \left ( g\circ f \right )&=g\left ( f(x) \right )\\ &=\left ( f(x) \right )^{2}+2\left ( f(x) \right )-3\\ &=\left ( x-1 \right )^{2}+2\left ( x-1 \right )-3\\ &=\left (x^{2}-2x+1 \right )+\left ( 2x-2 \right )-3\\ &=x^{2}-4 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 5.&\textrm{Diketahui fungsi}\: \: f(x)=6x-3\\ & \textrm{dan}\: \: g(x)=5x+4 \: \: \textrm{dan}\: \: (f\circ g)(a)=81\\ &\textrm{Nilai}\: \: a\: \: \textrm{adalah}\: ........\: \: (\textbf{Ebtanas 2001})\\ &\begin{array}{llllll}\\ \textrm{a}.&-2&&&\color{red}\textrm{d}.&2\\ \textrm{b}.&-1&\textrm{c}.&1&\textrm{e}.&27 \end{array}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Diketahui}\: \textrm{bahwa}:\\ &\begin{cases} &\color{blue}f(x) =6x-3\\ &\color{blue}g(x) =5x+4\end{cases}\\ &\left ( f\circ g \right )(a)=81\\ &\textrm{maka}\\ &(f\circ g)(x)=f\left ( g(x) \right )\\ &\: \qquad\qquad =6(g(x))-3\\ &\: \qquad\qquad =6(5x+4)-3\\ &\: \qquad\qquad =30x+24-3\\ &\: \qquad\qquad =30x+21=81\\ &30x+21=81\\ &30x=81-21=60\\ &x=\displaystyle \frac{60}{30}=2 \end{aligned} \end{array}$.