Question

Determine a função inversa das seguintes funções:
f) y=x/x-4
e) y=x³
d) y=3x-2/4x+3 (com x diferente de -3/4);

Answer 1

f)
$\displaystyle x=\frac{f^{-1}(x)}{f^{-1}(x)-4}\Longrightarrow x=1+\frac{4}{f^{-1}(x)-4} \\ \\ x=\frac{f^{-1}(x)}{f^{-1}(x)-4}\Longrightarrow x-1=\frac{4}{f^{-1}(x)-4} \\ \\ x=\frac{f^{-1}(x)}{f^{-1}(x)-4}\Longrightarrow f^{-1}(x)-4=\frac{4}{x-1} \\ \\ x=\frac{f^{-1}(x)}{f^{-1}(x)-4}\Longrightarrow f^{-1}(x)=4+\frac{4}{x-1} \\ \\ x=\frac{f^{-1}(x)}{f^{-1}(x)-4}\Longrightarrow \boxed{f^{-1}(x)=\frac{4x}{x-1}}$

e) 
$x=[f^{-1}(x)]^3\Longrightarrow \boxed{f^{-1}(x)=\sqrt[3]x}$

d)

$\displaystyle x=\frac{3f^{-1}(x)-2}{4f^{-1}(x)+3}\Longrightarrow x=\frac{3}{4}-\frac{17/4}{4f^{-1}(x)+3}\\ \\ x=\frac{3f^{-1}(x)-2}{4f^{-1}(x)+3}\Longrightarrow \frac{3}{4}-x=\frac{17/4}{4f^{-1}(x)+3}\\ \\ x=\frac{3f^{-1}(x)-2}{4f^{-1}(x)+3}\Longrightarrow 4f^{-1}(x)+3=\frac{17/4}{3/4-x}\\ \\ x=\frac{3f^{-1}(x)-2}{4f^{-1}(x)+3}\Longrightarrow 4f^{-1}(x)=\frac{17}{3-4x}-3\\ \\ x=\frac{3f^{-1}(x)-2}{4f^{-1}(x)+3}\Longrightarrow 4f^{-1}(x)=\frac{8+12x}{3-4x}$

$\displaystyle x=\frac{3f^{-1}(x)-2}{4f^{-1}(x)+3}\Longrightarrow \boxed{f^{-1}(x)=\frac{2+3x}{3-4x}}$

Answer 2

$f) f^{-1} :x = \frac{y}{y-4};y = xy-4x;y-xy=-4x; y(1-x)=-4x;y= \frac{-4x}{1-x};$ $e) f^{-1} :x = y^{3} --\ \textgreater \ y = \sqrt[3]{x} ;$ $d) f^{-1} :x = \frac{3y-2}{4y+3} --\ \textgreater \ 4xy+3x=3y-2--\ \textgreater \ 4xy-3y=-3x-2-->$ $4xy-3y=-3x-2--\ \textgreater \ y(4x-3)=-3x-2--\ \textgreater \ y= \frac{-3x-2}{4x-3}$