We are tasked with finding two functions $f(x)$ and $g(x)$ such that the given function $h(x) = (f \circ g)(x)$, which is the composition of $f(x)$ and $g(x)$.

The given function is: $h(x) = \frac{\sqrt{x - 5}}{x + 4}$

This means we need to express $h(x)$ as $f(g(x))$, where we will find appropriate functions for $f(x)$ and $g(x)$.

Let's examine the provided options:

Option A:

• $f(x) = \frac{1}{\sqrt{x}}$
• $g(x) = \sqrt{x + 4}$

Option B:

• $f(x) = \sqrt{x}$
• $g(x) = \frac{x - 5}{x + 4}$

Option C:

• $f(x) = \sqrt{x - 5}$
• $g(x) = \frac{1}{\sqrt{x + 4}}$

Option D:

• $f(x) = \frac{x - 5}{x + 4}$
• $g(x) = \sqrt{x}$

Now, let's break down $h(x)$ to find the correct functions $f(x)$ and $g(x)$.

Solution:

Looking at the form of $h(x) = \frac{\sqrt{x - 5}}{x + 4}$, it makes sense to set $g(x) = x - 5$, so that the square root operation can be applied easily to $f(g(x))$. However, since $h(x)$ involves both a square root in the numerator and a fraction structure, let's try Option D:

• $f(x) = \frac{x - 5}{x + 4}$
• $g(x) = \sqrt{x}$

The composition $f(g(x))$ gives: $f(g(x)) = f(\sqrt{x}) = \frac{\sqrt{x} - 5}{\sqrt{x} + 4}$ This does not match the given $h(x)$, so Option D is incorrect.

Let's evaluate Option B:

• $f(x) = \sqrt{x}$
• $g(x) = \frac{x - 5}{x + 4}$

The composition $f(g(x))$ gives: $f(g(x)) = f\left(\frac{x - 5}{x + 4}\right) = \sqrt{\frac{x - 5}{x + 4}}$ This matches the given $h(x)$, which is $\frac{\sqrt{x - 5}}{x + 4}$.

Thus, the correct answer is Option B.

Let me know if you would like more details on this solution or have further questions!

Five related questions:

1. What is the general process for finding $f(x)$ and $g(x)$ when given a composite function $h(x)$?
2. How does the composition of functions work?
3. Can you break down complex rational functions into simpler component functions?
4. How do transformations such as square roots and fractions interact in function compositions?
5. What is the domain of the function $h(x) = \frac{\sqrt{x - 5}}{x + 4}$?

Tip: When solving for composite functions, it helps to think of the composition as nested operations, where one function's output becomes the input for another function.