Decide whether or not the functions are inverses of each other.Use the definition of inverses to determine whether f and g are inverses.
-$f ( x ) = \sqrt { x + 2 } , x \geq - 2 ; \quad g ( x ) = x ^ { 2 } + 2$
A) $\mathrm { No }$
B) Yes

Question

Quizplus · Accepted Answer

For two functions to be inverses of each other, the composition of one with the other in both orders must yield the identity function. Testing $f(g(x)) = f(x^2 + 2) = \sqrt{x^2 + 2 + 2} = \sqrt{x^2 + 4}$, which does not simplify to $x$ for all $x$ in the domain of $g$. Similarly, $g(f(x)) = g(\sqrt{x + 2}) = (\sqrt{x + 2})^2 + 2 = x + 2 + 2 = x + 4$, which does not simplify to $x$ for all $x$ in the domain of $f$. Therefore, $f$ and $g$ are not inverses of each other.

Decide Whether or Not the Functions Are Inverses of Each f(x)=x+2,x≥−2;g(x)=x2+2f ( x ) = \sqrt { x + 2 } , x \geq - 2 ; \quad g ( x ) = x ^ { 2 } + 2f(x)=x+2​,x≥−2;g(x)=x2+2

Decide Whether or Not the Functions Are Inverses of Each $f ( x ) = \sqrt { x + 2 } , x \geq - 2 ; \quad g ( x ) = x ^ { 2 } + 2$