Use the definition of inverses to determine whether f and g are inverses.
-$$f ( x ) = \sqrt { x + 5 } , x \geq - 5 ; \quad g ( x ) = x ^ { 2 } + 5$$

Question

Quizplus · Accepted Answer

For two functions to be inverses, applying one after the other in any order should return the original input. However, if you apply $g$ after $f$, you get $g(f(x)) = (\sqrt{x + 5})^2 + 5 = x + 10$, not $x$. Similarly, applying $f$ after $g$ does not simplify to $x$ for all $x$ in the domain of $g$, due to the domain restriction on $f$ ($x \geq -5$) and the fact that $g(x) = x^2 + 5$ produces outputs that are always greater than or equal to 5, not accounting for the entire range of real numbers.

Use the Definition of Inverses to Determine Whether F and G