Decide whether the composite functions, f ° g and g ° 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

For $f \circ g(x)$, we substitute $g(x)$ into $f(x)$, getting $f(g(x)) = (\sqrt{x} - 1)^2 + 1 = x - 2\sqrt{x} + 2$, which is not equal to $x$. For $g \circ f(x)$, we substitute $f(x)$ into $g(x)$, getting $g(f(x)) = \sqrt{x^2 + 1} - 1$, which is also not equal to $x$. Therefore, neither composite function equals $x$.

Decide Whether the Composite Functions, F ° G and G f(x)=x2+1,g(x)=x−1f ( x ) = x ^ { 2 } + 1 , g ( x ) = \sqrt { x } - 1f(x)=x2+1,g(x)=x​−1

Decide Whether the Composite Functions, F ° G and G $f ( x ) = x ^ { 2 } + 1 , g ( x ) = \sqrt { x } - 1$