Diketahui $f(x) = \frac {x+7}{5x-2}, x\ne \frac {2}{5} $ maka $f^{-1}(x) = ... $
A. $\frac {2x+7}{5x-2}, x \ne \frac {1}{5} $
B. $\frac {5x-2}{x+7}, x \ne -7 $
C. $\frac {7x+1}{-2x+5}, x \ne \frac{5}{2} $
D. $\frac {x-2}{5x+7}, x \ne -\frac {7}{5} $
E. $\frac {2x-7}{5x+1}, x \ne -\frac {1}{5} $
Penyelesaian
Diketahui:
$f(x) = \frac {x+7}{5x-2}$
Ditanya:
$f^{-1}(x)$
Jawab
Cara Manual
misal $f(x) = y = \frac {x+7}{5x-2}$
$y=\frac {x+7}{5x-2} $
$y(5x-2)=x+7 $
$5xy-2y=x+7 $
$5xy-x=2y+7 $
$x(5y-1)=2y+7$
$x=\frac {2y+7}{5y-1} $
$f^{-1}(y)= \frac {2y+7}{5y-1} $
$f^{-1}(x)= \frac {2x+7}{5x-1} $
Cara Cepat
$f(x)=\frac {ax+b}{cx+d} \to f^{-1}(x)= \frac {-dx+b}{cx-a} $
$f(x) = \frac {x+7}{5x-2}$
$a=1$
$b=7$
$c=5$
$d=-2$
$f^{-1}(x)= \frac {2x+7}{5x-1} $
Kesimpulan
Jadi, nilai inversnya adalah $f^{-1}(x)= \frac {2x+7}{5x-1} $
Muda Berkarya Intelektual Normatif