Determine whether (f ∘ g)(x) = x and whether (g ∘ f)(x) = x.
-$$f ( x ) = \sqrt { x + 1 } , g ( x ) = x ^ { 2 }$$
A) No, no
B) Yes, yes
C) Yes, no
D) No, yes

Question

Quizplus · Accepted Answer

For $(f \circ g)(x) = f(g(x)) = f(x^2) = \sqrt{x^2 + 1}$, which is not equal to $x$ because of the added 1 inside the square root. For $(g \circ f)(x) = g(f(x)) = g(\sqrt{x + 1}) = (\sqrt{x + 1})^2 = x + 1$, which is also not equal to $x$ because of the added 1. Therefore, both compositions do not return $x$.

Determine Whether (F ∘ G)(x) = X and Whether (G f(x)=x+1,g(x)=x2f ( x ) = \sqrt { x + 1 } , g ( x ) = x ^ { 2 }f(x)=x+1​,g(x)=x2

Determine Whether (F ∘ G)(x) = X and Whether (G $f ( x ) = \sqrt { x + 1 } , g ( x ) = x ^ { 2 }$