Contoh Soal 6 Fungsi Eksponen (Matematika Peminatan Kelas X)

$\begin{array}{l}\\ 26.& (\mathbf{\text{UM UGM 05}})\text{Hasil dari}\\ & \sqrt{0,3+\sqrt{0,08}}=\sqrt{a}+\sqrt{b},\text{maka}{\displaystyle \frac{1}{a}+\frac{1}{b}=....}\\ & \begin{array}{l}\\ \text{a}.& 25\\ \text{b}.& 20\\ \text{c}.& 15\\ \text{d}.& 10\\ \text{e}.& 5\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}\sqrt{0,3+\sqrt{0,08}}& =\sqrt{0,2+0,1+\sqrt{4\times 0,2\times 0,1}}\\ & =\sqrt{0,2+0,1+2\sqrt{\times 0,2\times 0,1}}\\ & =\sqrt{0,2}+\sqrt{0,1}\\ \text{maka},a=0,2& ,b=0,1\\ \text{sehingga}{\displaystyle \frac{1}{a}+\frac{1}{b}}& ={\displaystyle \frac{1}{0,2}+\frac{1}{0,1}=5+10=15}\end{array}\end{array}$

$\begin{array}{l}\\ 27.& (\mathbf{\text{SPMB 06}})\text{Jika bilangan bulat}a\text{dan}b\text{memenuhi}\\ & {\displaystyle \frac{\sqrt{5}-\sqrt{6}}{\sqrt{5}+\sqrt{6}}=a+b\sqrt{30},\text{maka}ab=....}\\ & \begin{array}{l}\\ \text{a}.& -22\\ \text{b}.& -11\\ \text{c}.& -9\\ \text{d}.& 2\\ \text{e}.& 13\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}{\displaystyle \frac{\sqrt{5}-\sqrt{6}}{\sqrt{5}+\sqrt{6}}}& ={\displaystyle \frac{\sqrt{5}-\sqrt{6}}{\sqrt{5}+\sqrt{6}}\times {\displaystyle \frac{\sqrt{5}-\sqrt{6}}{\sqrt{5}-\sqrt{6}}}}\\ & ={\displaystyle \frac{5-2\sqrt{30}+6}{5-6}}\\ & ={\displaystyle \frac{11-2\sqrt{30}}{-1}}\\ & =-11+2\sqrt{30}\\ \text{sehingga}& a=-11,b=2,\text{maka}\\ ab& =(-11)\times 2\\ & =-22\end{array}\end{array}$

$\begin{array}{l}\\ 28.& (\mathbf{\text{OSK 2013}})\text{Misal}a\text{dan}b\text{bilangan asli}\\ & \text{dengan}a>b.\text{Jika}\sqrt{94+2\sqrt{2013}}=\sqrt{a}+\sqrt{b}\\ & \text{maka nilai}a-b\text{adalah... .}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}\sqrt{94+2\sqrt{2013}}& =\sqrt{61+33+2\sqrt{61\times 33}}\\ & =\sqrt{61}+\sqrt{33}\\ & =\sqrt{a}+\sqrt{b}\\ \text{Sehingga}a& =61,b=33,\text{maka}\\ a-b& =61-33\\ & =28\end{array}\end{array}$

$\begin{array}{l}\\ 29.& \text{Daerah hasil dari fungsi eksponen}y=x^{-\frac{2}{3}}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& y<0\\ \text{b}.& y>0\\ \text{c}.& y\ge 0\\ \text{d}.& y\le 0\\ \text{e}.& \text{Semua bilangan real}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \text{Perhatikanlah gambar berikut}\end{array}$

$.\begin{array}{rl}\text{diketahui}& \\ y& =x^{-{\displaystyle \frac{2}{3}}}\\ y^3& =x^{-2}\\ y^3& ={\displaystyle \frac{1}{x^2},\text{atau}}\\ y^3\times x^2& =1,\\ \text{sehingga}& y\text{tidak mungkin berharga}0\end{array}$

$\begin{array}{l}\\ 30.& \text{Jika}f(x)=b^x,\text{di mana konstan positif},\\ \\ & {\displaystyle \frac{f(x^2+x)}{f(x+1)}=....}\\ & \begin{array}{l}\\ \text{a}.& f\left(x^2\right)& & & \\ \text{b}.& f(x+1)f(x-1)\\ \text{c}.& f(x+1)+f(x-1)\\ \text{d}.& f(x+1)-f(x-1)\\ \text{e}.& f(x^2-1)\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}{\displaystyle \frac{f(x^2+x)}{f(x+1)}}& =\frac{b^{x^2+x}}{b^{x+1}}\\ & =b^{x^2+x-(x+1)}\\ & =b^{x^2-1}\\ & =f(x^2-1)\end{array}\end{array}$