Question

Dadas as funções f e g, dettermine a função inversa de gof:
a- f(x)=4x+1 e g(x)=3x-6

b-f(x)=x²-1 e g(x)= \sqrt{x} +4

Answer 1

a) (gof)(x) = g[f(x)] = g(4x+1) = 3(4x+1)-6 = 12x-3

(gof)(x) = 12x - 3

despejamos x

$x=\frac{(g\circ f)(x) +3}{12}\Longrightarrow (g\circ f)^{-1}(x)=\frac{x+3}{12}$

b)  $(g\circ f)(x) = g[f(x)]=g(x^2-1)=\sqrt{x^2-1}+4$, es decir

$(g\circ f)(x) =\sqrt{x^2-1}+4$

despejemos x

$(g\circ f)(x) = \sqrt{x^2-1}+4\Longrightarrow (g\circ f)(x)-4 =\sqrt{x^2-1}\\ \\ (g\circ f)(x) = \sqrt{x^2-1}+4\Longrightarrow [(g\circ f)(x)-4]^2 =x^2-1\\ \\ (g\circ f)(x) = \sqrt{x^2-1}+4\Longrightarrow [(g\circ f)(x)-4]^2 +1=x^2\\ \\ (g\circ f)(x) = \sqrt{x^2-1}+4\Longrightarrow \sqrt{[(g\circ f)(x)-4]^2 +1}=x\\ \\ \\ (g\circ f)^{-1}(x) =\sqrt{(x-4)^2 +1}$