Jika $f(x)=3x-2 $ dan $g(x)=\frac {x+2}{x-5}$, fungsi $(g \circ f)^{-1}(x)=...$
a. $\frac {3x}{3x-7} $
b. $\frac {3x}{3x+7} $
c. $\frac {3x-7}{3x} $
d. $\frac {3x-3}{7x} $
e. $\frac {7x}{3x-3} $
Pembahasan
Diketahui:
$f(x)=3x-2$
$g(x)=\frac {x+2}{x-5} $
Ditanya:
$(g\circ f)^{-1}(x) =...$
jawab
mencari nilai $(g\circ f)(x)$ terlebih dahulu
$(g\circ f)(x)=(g(3x-2)) $
$(g\circ f)(x)= \frac {(3x-2)+2}{(3x-2)-5}$
$(g\circ f)(x)=\frac {3x}{3x-7} $
cara manual
misal $(g\circ f)(x)=y=\frac {3x}{3x-7} $
$y=\frac {3x}{3x+7} $
$y(3x-7)=3x$
$3xy-7y=3x$
$3xy-3x=7y $
$x(3y-3)=7y$
$x=\frac {7y}{3y-3} $
$(g\circ f)^{-1}(y)=\frac {7y}{3y-3} $
$(g\circ f)^{-1}(x)=\frac {7x}{3x-3} $
cara cepat
$f(x) = \frac {ax+b}{cx+d}\to f^{-1}(x)=\frac {-dx+b}{cx-a} $
$a=3$
$b=0$
$c=3$
$d=-7$
$(g\circ f)(x)= \frac {3x}{3x-7}$
$(g\circ f)^{-1}(x)= \frac {7x}{3x-3}$
Kesimpulan
Jadi, fungsi inversnya adalah $(g\circ f)^{-1}(x)= \frac {7x}{3x-3}$
Muda Berkarya Intelektual Normatif