Question

2)Determine a função inversa das seguintes funções:​

Answer 1

$\large\boxed{\begin{array}{l}\tt a)~\sf f(x)=7-\dfrac{3x}{5}\\\sf 5f(x)=35-3x\\\sf 3x=35-5f(x)\\\sf x=\dfrac{35-5f(x)}{3}\\\\\sf f^{-1}(x)=\dfrac{35-5x}{3}\end{array}}$

$\large\boxed{\begin{array}{l}\tt b)~\sf f(x)= 8-\dfrac{2x}{3}\\\sf 3f(x)=24-2x\\\sf 2x=24-3f(x)\\\sf x=\dfrac{24-3f(x)}{2}\\\\\sf f^{-1}(x)=\dfrac{24-3x}{2}\end{array}}$

$\large\boxed{\begin{array}{l}\tt c)~\sf f(x)=5x-\dfrac{2}{4}\\\sf 4f(x)=20x-2\\\sf\\\sf 20x=4f(x)+2\\\sf x=\dfrac{4f(x)+2}{20}\\\\\sf f^{-1}(x)=\dfrac{4x+2}{20}\end{array}}$

$\large\boxed{\begin{array}{l}\tt d)~\sf f(x)=4x-\dfrac{3}{6}\\\sf 6f(x)=24x-3\\\sf 24x= 6f(x)+3\\\sf x=\dfrac{6f(x)+3}{24}\\\\\sf f^{-1}(x)=\dfrac{6x+3}{24}\end{array}}$

$\large\boxed{\begin{array}{l}\tt e)~\sf f(x)=\sqrt[\sf3]{\sf 5x-3}\\\sf f(x)^3=5x-3\\\sf 5x= f(x)^3+3\\\sf x=\dfrac{f(x)^3+3}{5}\\\\\sf f^{-1}(x)=\dfrac{x^3+3}{5}\end{array}}$

$\large\boxed{\begin{array}{l}\tt f)~\sf f(x)=\sqrt[\sf4]{\sf 6x-2}\\\sf f(x)^4=6x-2\\\sf 6x=f(x)^4+2\\\sf x=\dfrac{f(x)^4+2}{6}\\\\\sf f^{-1}(x)=\dfrac{x^4+2}{6}\end{array}}$

$\large\boxed{\begin{array}{l}\tt g )~\sf f(x)=\dfrac{3x-5}{4x-9}\\\sf 4xf(x)-9f(x)=3x-5\\\sf 4xf(x)-3x=9f(x)-5\\\sf x\cdot(4f(x)-3)=9f(x)-5\\\sf x=\dfrac{9f(x)-5}{4f(x)-3}\\\\\sf f^{-1}(x)=\dfrac{9x-5}{4x-3}\end{array}}$

$\large\boxed{\begin{array}{l}\tt h)~\sf f(x)=\dfrac{2x-6}{3x-7}\\\\\sf 3xf(x)-7f(x)=2x-6\\\sf 3xf(x)-2x=7f(x)-6\\\sf x\cdot(3f(x)-2)=7f(x)-6\\\sf x=\dfrac{7f(x)-6}{3f(x)-2}\\\\\sf f^{-1}(x)=\dfrac{7x-6}{3x-2}\end{array}}$