Decide whether the composite functions, f $$f \circ g$$ nd $\mathbf { g } \circ \mathrm { f }$ f, are equal to x.
-$f ( x ) = x ^ { 2 } + 1 , g ( x ) = \sqrt { x } - 1$
A) No, yes
B) No, no
C) Yes, yes
D) Yes, no

Question

Quizplus · Accepted Answer

To determine if $f \circ g$ and $g \circ f$ are equal to $x$, we need to compute each composite function and compare it to $x$.1. For $f \circ g$, we substitute $g(x)$ into $f(x)$:$$f(g(x)) = (\sqrt{x} - 1)^2 + 1 = x - 2\sqrt{x} + 1 + 1 = x - 2\sqrt{x} + 2$$This is not equal to $x$.2. For $g \circ f$, we substitute $f(x)$ into $g(x)$:$$g(f(x)) = \sqrt{x^2 + 1} - 1$$This is also not equal to $x$.Therefore, neither $f \circ g$ nor $g \circ f$ is equal to $x$.

Decide Whether the Composite Functions, F f∘gf \circ gf∘g Nd g∘f\mathbf { g } \circ \mathrm { f }g∘f

Decide Whether the Composite Functions, F $f \circ g$ Nd $\mathbf { g } \circ \mathrm { f }$