Question

Sendo f(x)=-10-2x/-11+x
Calcular o valor a função inversa de 2/3

Answer 1

$f(x)=\dfrac{-10-2x}{-11+x}$

Substituindo $x$ por $y$ e $f(x)$ por $x$:

$x=\dfrac{-10-2y}{-11+y} \iff -11x+xy=-10-2y \iff xy+2y=11x-10$

$y(x+2)=11x-10 \iff y=\dfrac{11x-10}{x+2}$

Logo, $f^{-1}\left(\dfrac{2}{3}\right)$ vale:

$f^{-1}\left(\dfrac{2}{3}\right)=\dfrac{11\cdot\dfrac{2}{3}-10}{\dfrac{2}{3}+2}=\dfrac{\dfrac{22}{3}-10}{\dfrac{8}{3}}=\dfrac{-\dfrac{8}{3}}{\dfrac{8}{3}}=\dfrac{-8}{3}\cdot\dfrac{3}{8}=\dfrac{-24}{24}$

$\boxed{f^{-1}\left(\dfrac{2}{3}\right)=-1}$