Diketahui fungsi $f(x)=\frac{x+2}{x-5}$ dan $g(x)=2x+3$, maka $(g\circ f)^{-1}(x)=$...
A. $\frac{5x+11}{x-5}$
B. $\frac{5x-1}{x+11}$
C. $\frac{5x-11}{x-5}$
D. $\frac{x-5}{5x-11}$
E. $\frac{x+5}{5x-11}$
Diketahui:
$f(x)=\frac{x+2}{x-5}$
$g(x)=2x+3$
Ditanya: $(g\circ f)^{-1}(x)=$
Jawab
$(g\circ f)(x)=2(\frac{x+2}{x-5})+3$
$(g\circ f)(x)=\frac{2x+4}{x-5}+3$
$(g\circ f)(x)=\frac{2x+4}{x-5}+\frac{3(x-5)}{x-5}$
$(g\circ f)(x)=\frac{2x+4+3x-15}{x-5}$
$(g\circ f)(x)=\frac{5x-11}{x-5}$
Cara 1
misal $(g\circ f)(x)=y$
$\frac{5x-11}{x-5}=y$
$(5x-11)=y(x-5)$
$5x-11=yx-5y$
$5y-11=yx-5x$
$5y-11=x(y-5)$
$\frac{5y-11}{y-5}=x$
$(g\circ f)^{-1}(x)=\frac{5x-11}{x-5}$
Cara 2
$f(x)=\frac{ax+b}{cx+d}\to f^{-1}(x)=\frac{-dx+b}{cx-a}$
$(g\circ f)(x)=\frac{5x-11}{x-5}$
$(g\circ f)^{-1}(x)=\frac{5x-11}{x-5}$
Muda Berkarya Intelektual Normatif