Contoh Soal 2 Fungsi

$\begin{array}{ll}\\ 6.&\textrm{Diketahui bahwa}\\ &f(x)=\begin{cases} 0,&\textrm{untuk}\: x<0\\ x^{2},&\textrm{untuk}\: 0\leq x< 1\\ 2x-1,&\textrm{untuk}\: x\geq 1 \end{cases}\\ &\textrm{Nilai dari}\: f(-1)+f\left ( \displaystyle \frac{1}{2} \right )-f(3)\: \textrm{adalah}.... \\ &\begin{array}{llllll}\\ \textrm{a}.&-5\displaystyle \frac{1}{4}&&&\textrm{d}.&4\displaystyle \frac{3}{4}\\\\ \color{red}\textrm{b}.&-4\displaystyle \frac{3}{4}&\textrm{c}.&4&\textrm{e}.&5\displaystyle \frac{1}{4} \end{array}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{Diketahui}\: &\\ f(x)&=\begin{cases} 0,&\textrm{untuk}\: x<0\\ x^{2},&\textrm{untuk}\: 0\leq x< 1\\ 2x-1,&\textrm{untuk}\: x\geq 1 \end{cases}\\ \textrm{maka nilai}&\: \: f(-1)+f\left ( \displaystyle \frac{1}{2} \right )-f(3)\\ &=0+\left ( \displaystyle \frac{1}{2} \right )^{2}-\left ( 2(3)-1 \right )\\ &=\displaystyle \frac{1}{4}-5\\ &=-4\displaystyle \frac{3}{4} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 7.&\textrm{Jika diketahui}\: \: f\left ( x+\displaystyle \frac{1}{x} \right )=x^{3}+\displaystyle \frac{1}{x^{3}}\: ,\\ &\textrm{maka nilai dari}\: \: f\left ( \displaystyle \frac{5}{2} \right )\: \: \textrm{adalah}.... \\ &\begin{array}{llllll}\\ \textrm{a}.&2\displaystyle \frac{1}{8}&&&\color{red}\textrm{d}.&8\displaystyle \frac{1}{8}\\\\ \textrm{b}.&2\displaystyle \frac{1}{2}&\textrm{c}.&4\displaystyle \frac{1}{8}&\textrm{e}.&12\displaystyle \frac{1}{8} \end{array}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{Perhatikan}\, &\: \textrm{bahwa}:\\ (p+q)^{3}&=p^{3}+3p^{2}q+3pq^{2}+q^{3}\\ \textrm{Jika kita su}&\textrm{bstitusikan}\: \: p=x^{3}\: \: \textrm{dan}\: \: q=\displaystyle \frac{1}{x^{3}}\\ \left ( x+\displaystyle \frac{1}{x} \right )^{3}&=x^{3}+3x^{2}\left ( \displaystyle \frac{1}{x} \right )+3x\left ( \displaystyle \frac{1}{x} \right )^{2}+\left ( \displaystyle \frac{1}{x} \right )^{3}\\ &=x^{3}+\left ( \displaystyle \frac{1}{x} \right )^{3}+3x+\displaystyle \frac{3}{x}\\ &=\left ( x^{3}+\displaystyle \frac{1}{x^{3}} \right )+3\left ( x+\displaystyle \frac{1}{x} \right )\\ \textrm{sehingga}\quad&\\ f\left ( x+\displaystyle \frac{1}{x} \right )&=x^{3}+\displaystyle \frac{1}{x^{3}}\\ &=\left ( x+\displaystyle \frac{1}{x} \right )^{3}-3\left ( x+\displaystyle \frac{1}{x} \right )\\ f(u)&=u^{3}-3u,\: \: \textrm{maka}\\ f\left ( \displaystyle \frac{5}{2} \right )&=\left ( \displaystyle \frac{5}{2} \right )^{3}-3\left ( \displaystyle \frac{5}{2} \right )\\ &=\displaystyle \frac{125}{8}-\frac{15}{2}\\ &=\displaystyle \frac{65}{8}\\ &=8\displaystyle \frac{1}{8} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 8.&\textrm{Misal fungsi}\: \: f\: \: \textrm{terdefinisi untuk }\\ &\textrm{seluruh bilangan real}\: \: x.\\ &\textrm{Jika}\: \: f(p+q)=f(pq)\: \: \textrm{untuk semua}\: \: p,\: q\\ & \textrm{bilangan bulat positif dan}\: \: f(1)=2,\\ &\textrm{maka nilai}\: \: f(2021)=....\\ &\begin{array}{llllll}\\ \textrm{a}.&0&&&\textrm{d}.&3\\\\ \textrm{b}.&1&\color{red}\textrm{c}.&2&\textrm{e}.&5 \end{array}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{Diketahui}\: & \textrm{bahwa}\\ f(1)&=2\: \: \textrm{dan}\\ f(p+q)&=f(pq)\\ \textrm{maka}&\\ f(2)&=f(1+1)=f(1.1)=f(1)=2\\ f(3)&=f(1+2)=f(1.2)=f(2)=f(1)=2\\ f(4)&=f(1+3)=f(1.3)=f(3)=f(2)=f(1)=2\\ f(5)&=f(1+4)=f(1.4)=f(4)=f(3)=f(2)=f(1)=2\\ \vdots &\\ f(2021)&=\cdots =\cdots =\cdots =f(2)=f(1)=2 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 9.&\textrm{Jika}\: \: a_{_{0}}=\displaystyle \frac{2}{5}\: \: \textrm{dan}\: \: a_{n+1}=2\left | a_{n} \right |-1, \textrm{maka nilai}\: \: a_{_{2022}}\\ &\textrm{adalah}....\\ &\begin{array}{llllll}\\ \color{red}\textrm{a}.&-0,6&&&\textrm{d}.&0,4\\\\ \textrm{b}.&-0,2&\textrm{c}.&0,2&\textrm{e}.&0,6 \end{array}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{Diketahui}\: & \textrm{bahwa}\\ a_{_{0}}&=\displaystyle \frac{2}{5}=0,4\: \: \textrm{dan}\: \: a_{n+1}=2\left | a_{n} \right |-1,\: \textrm{maka}\\ a_{_{1}}&=2\left | a_{_{0}} \right |-1=2\left | 0,4 \right |-1=0,8-1=-0,2\\ a_{_{2}}&=2\left | a_{_{1}} \right |-1=2\left | -0,2 \right |-1=2(0,2)-1\\ &=0,4-1=-0,6\\ a_{_{3}}&=2\left | a_{_{2}} \right |-1=2\left | -0,6 \right |-1=1,2-1=0,2\\ a_{_{4}}&=2\left | a_{_{3}} \right |-1=2\left | 0,2 \right |-1=0,4-1=-0,6=\textbf{a}_{_{\textbf{2}}}\\ a_{_{5}}&=2\left | a_{_{4}} \right |-1=2\left | -0,6 \right |-1=1,2-1=0,2=\textbf{a}_{_{\textbf{3}}}\\ a_{_{6}}&=2\left | a_{_{5}} \right |-1=2\left | 0,2 \right |-1=0,4-1=-0,6=\textbf{a}_{_{\textbf{2}}}\\ a_{_{7}}&=2\left | a_{_{6}} \right |-1=2\left | -0,6 \right |-1=1,2-1=0,2=\textbf{a}_{_{\textbf{3}}}\\ \vdots &\\ a_{_{2022}}&=\cdots =\cdots =\textbf{a}_{_{\textbf{2}}}=-0,6 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 10.&\textrm{Kurva}\: \: f(x)=\displaystyle \frac{10}{x^{2}-10x+25}\\ &\textrm{mempunyai asimtot vertikal pada}....\\ &\begin{array}{llllll}\\ \textrm{a}.&x=0\: \: \textrm{saja}&&&\\ \color{red}\textrm{b}.&x=5\: \: \textrm{saja}&\\ \textrm{c}.&x=10\: \: \textrm{saja}\\ \textrm{d}.&x=0\: \: \textrm{dan}\: \: x=5\: \: \textrm{saja}\\ \textrm{e}.&x=0,\: x=5,\: \: \textrm{dan}\: \: x=10 \end{array}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textbf{Asimtot vertikal}&(\textbf{tegak})\: \textrm{diperoleh saat}\\ x^{2}-10x+25&=0\\ (x-5)^{2}&=0\\ x-5&=0\\ x&=5\\\\ \textrm{Ilustrasinya gamba}&\textrm{rnya adalah sebagai berikut}: \end{aligned} \end{array}.$