To determine if $f(x)$ is the inverse of $g(x)$, we must verify if applying one function to the other returns the input $x$. That is, we need to check the following two conditions:

1. $f(g(x)) = x$
2. $g(f(x)) = x$

Given:

• $f(x) = (x + 9)^2$, where $x \geq -9$
• $g(x) = \sqrt{x - 9}$, defined for $x \geq 9$

Step 1: Verify $f(g(x)) = x$

Substitute $g(x)$ into $f(x)$:

$f(g(x)) = \left( \sqrt{x - 9} + 9 \right)^2$

Simplify:

$f(g(x)) = \left( \sqrt{x - 9} + 9 \right)^2 = (\sqrt{x - 9})^2 = x - 9 + 9 = x$

Thus, $f(g(x)) = x$, which satisfies the first condition.

Step 2: Verify $g(f(x)) = x$

Now substitute $f(x)$ into $g(x)$:

$g(f(x)) = \sqrt{(x + 9)^2 - 9}$

Simplify:

$g(f(x)) = \sqrt{(x + 9)^2 - 9} = \sqrt{x^2 + 18x + 81 - 9} = \sqrt{x^2 + 18x + 72}$

This does not simplify to $x$. Therefore, $g(f(x)) \neq x$, meaning that the second condition fails.

Conclusion:

Since $g(f(x)) \neq x$, $f(x)$ is not the inverse of $g(x)$. The correct answer is no.

Would you like more details on any part of this explanation?

Here are five related questions for further exploration:

1. How do you find the inverse of a function algebraically?
2. What are the domain and range requirements for inverse functions?
3. Can every function have an inverse?
4. How can you graphically determine if two functions are inverses of each other?
5. What are some common pairs of inverse functions?

Tip: Always check both $f(g(x))$ and $g(f(x))$ to ensure functions are inverses!