Question

Determine a inversa de cada função bigetiva.

Answer 1

Explicação passo-a-passo:

a) $f(x)=-4x+1$

$y=-4x+1$

Trocamos $x$ por $y$ e $y$ por $x$

$x=-4y+1$

Isolamos $y$:

$4y=-x+1$

$y=\dfrac{-x+1}{4}$

$f^{-1}(x)=\dfrac{-x+1}{4}$

b) $f(x)=\sqrt[5]{x+3}$

$y=\sqrt[5]{x+3}$

$x=\sqrt[5]{y+3}$

$x^5=y+3$

$y=x^5-3$

$f^{-1}(x)=x^5-3$

c) $f(x)=\dfrac{x}{2}-5$

$y=\dfrac{x}{2}-5$

$x=\dfrac{y}{2}-5$

$2x=y-10$

$y=2x+10$

$f^{-1}(x)=2x+10$

d) $f(x)=\dfrac{2}{x-1}$

$y=\dfrac{2}{x-1}$

$x=\dfrac{2}{y-1}$

$x\cdot(y-1)=2$

$xy-x=2$

$xy=x+2$

$y=\dfrac{x+2}{x}$

$f^{-1}(x)=\dfrac{x+2}{x}$, com $x\ne0$

e) $f(x)=3x^2+4$

$y=3x^2+4$

$x=3y^2+4$

$3y^2=x-4$

$y^2=\dfrac{x-4}{3}$

$y=\sqrt{\dfrac{x-4}{3}}$

$f^{-1}(x)=\sqrt{\dfrac{x-4}{3}}$, com $x\ge4$

f) $f(x)=\dfrac{6x-1}{3x+2}$

$y=\dfrac{6x-1}{3x+2}$

$x=\dfrac{6y-1}{3y+2}$

$6y-1=3xy+2x$

$6y-3xy=2x+1$

$y\cdot(6-3x)=2x+1$

$y=\dfrac{2x+1}{6-3x}$

$f^{-1}(x)=\dfrac{2x+1}{6-3x}$, com $x\ne2$