Question

Determine para cada lei que define a função, as compostas de g º f:


1) f(x) = 2x+1/x-2, g(x) = x;

2) f(x) = 3, g(x) = x + 3

3) f(x) = 1/x-2, g(x) = 2x-4

Answer 1

Olá!
 
       Lembre que   $(g\circ f )(x)=g(f(x)).$


1)


$f(x) = \dfrac{2x+1}{x-2},\;\; g(x) = x\Rightarrow (g\circ f)(x)=g(f(x)) = \\ \\ \\ g\left(\dfrac{2x+1}{x-2}\right) = \dfrac{2x+1}{x-2},\;\;\forall x\neq 2.$


2)


$f(x) = 3,\;\; g(x) = x + 3\Rightarrow (g\circ f)(x)=g(f(x))=\\ \\ g(3)=3+3=6.$


3)


$f(x) = \dfrac{1}{x-2},\;\; g(x) = 2x-4\Rightarrow (g\circ f)(x)=g(f(x))=\\ \\ \\ = g\left(\dfrac{1}{x-2}\right)=2\cdot\left(\dfrac{1}{x-2}\right)-4=\dfrac{2}{x-2}-4=\\ \\ \\ = \dfrac{2-4(x-2)}{x-2}=\dfrac{2-4x+4}{x-2}=\dfrac{-4x+6}{x-2},\;\;\forall x\neq 2.$



Bons estudos!