Jika f(x)=8-2x dan g(x)=x+ 1/x-3 maka tentukan g o g^-1 dan f^-1 o f

Jika f(x)=8-2x dan g(x)=x+ 1/x-3 maka tentukan g o g^-1 dan f^-1 o f

Jawaban Terkonfirmasi

G(x) = $frac{x+1}{x-3}$
y = $frac{x+1}{x-3}$
y(x-3) = x+1
xy-3y = x+1
xy-x = 3y+1
x (y-1) = 3y+1
x = $frac{3y+1}{y-1}$
$g^{-1} (x)$ = $frac{3x+1}{x-1}$

shgga (g o $g^{-1}$ ) (x) = g ( $g^{-1}(x))$
= g ( $frac{3x+1}{x-1}$ )
= $frac{ frac{3x+1}{x-1} + 1 }{ frac{3x+1}{x-1} -3 } }$
—————————————– x $frac{x-1}{x-1}$
= $frac{3x+1 + x-1}{3x+1-3x+3}$
= $frac{4x}{4} = 4$

f(x) = 8-2x
y = 8-2x
2x = 8-y
x = $frac{8-y}{2}$
$f^{-1} (x) = frac{8-x}{2}$

Shgga $f^{-1} o f$ = $f^{-1} (f(x)) = f^{-1}(8-2x)$
= $frac{8-(8-2x)}{2}$
= $frac{2x}{2} = 2$