Question

Considerando as funções f: R ⟶ R, g: R ⟶ R definidas por f(x) = 2x + 1, g(x) = x2 – x + 2 e h(x) = 3 – x, determine:

a) f(g(x))

b) g(f(x))

c) g(f(h(x)))​

Answer 1

$\large\boxed{\begin{array}{l}\sf f(x)=2x+1~~g(x)=x^2-x+2~~h(x)=3-x\\\tt a)~\sf f(g(x))=2g(x)+1\\\sf f(g(x))=2\cdot(x^2-x+2)+1\\\sf f(g(x))=2x^2-2x+4+1\\\sf f(g(x))=2x^2-2x+5\end{array}}$

$\large\boxed{\begin{array}{l}\tt b)~\sf g(f(x))=f(x)^2-f(x)+2\\\sf g(f(x))=(2x+1)^2-(2x+1)+2\\\sf g(f(x))=4x^2+4x+\backslash\!\!\!1-2x-\backslash\!\!\!1+2\\\sf g(f(x))=4x^2+2x+2\end{array}}$

$\large\boxed{\begin{array}{l}\tt c)~\sf f(h(x))= 2h(x)+1\\\sf f(h(x))= 2\cdot(3-x)+1\\\sf f(h(x))=6-2x+1\\\sf f(h(x))=-2x+7\\\sf g(f(h(x)))=(f(h(x)))^2-f(h(x))+2\\\sf g(f(h(x)))=(-2x+7)^2-(-2x+7)+2\\\sf g(f(h(x)))=4x^2-28x+49+2x-7+2\\\sf g(f(h(x)))=4x^2-26x+44\end{array}}$