Question

2- Dada as funçoes f(x)= √x-5 e g(x)= x+4, determina f o g(r), g o f(r) e suas inversas

Answer 1

$f(x)=\sqrt{x} -5\\g(x)=x+4\\f o g(r) = f[g(r)] => g(r) = r + 4\\f(r+4)=\sqrt{r+4}-5\\\\gof=g[f(r)] => f(r) = \sqrt{r} -5\\g(\sqrt{r}-5)=\sqrt{r}-5+4=\sqrt{r}-1$

Inversa de f(x):

$y=\sqrt{x} -5\\x=\sqrt{y} -5\\-\sqrt{y}=-x-5\\\sqrt{y} =x+5\\(\sqrt{y} )^{2} =(x+5)^{2} \\y=(x+5)(x+5)\\y=x^{2} +5x+5x+25\\y^{-1} =x^{2} +10x+25$

Inversa de g(x):

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