Determine whether (f ∘ g)(x) = x and whether (g ∘ f)(x) = x.
-$$f ( x ) = x ^ { 2 } + 2 , g ( x ) = \sqrt { x } - 2$$
A) Yes, no
B) No, yes
C) Yes, yes
D) No, no

Question

Quizplus · Accepted Answer

To determine if these compositions equal $x$, we compute both $(f \circ g)(x)$ and $(g \circ f)(x)$.First, $(f \circ g)(x) = f(g(x)) = f(\sqrt{x} - 2) = (\sqrt{x} - 2)^2 + 2 = x - 4\sqrt{x} + 4 + 2 = x - 4\sqrt{x} + 6$, which is not equal to $x$.Second, $(g \circ f)(x) = g(f(x)) = g(x^2 + 2) = \sqrt{x^2 + 2} - 2$, which is also not equal to $x$.Therefore, neither composition equals $x$.

Determine Whether (F ∘ G)(x) = X and Whether (G f(x)=x2+2,g(x)=x−2f ( x ) = x ^ { 2 } + 2 , g ( x ) = \sqrt { x } - 2f(x)=x2+2,g(x)=x​−2

Determine Whether (F ∘ G)(x) = X and Whether (G $f ( x ) = x ^ { 2 } + 2 , g ( x ) = \sqrt { x } - 2$