Let's first define the functions you gave:

• $f(x) = \sqrt{x} + 8$
• $g(x) = \frac{1}{x} - 8$

Now, let's compute the two composite functions $f \circ g$ and $g \circ f$.

1. $(f \circ g)(x) = f(g(x))$

This means we need to substitute $g(x)$ into $f(x)$.

From the definition of $f(x)$, we have: $f(x) = \sqrt{x} + 8$

To compute $f(g(x))$, substitute $g(x) = \frac{1}{x} - 8$ into $f(x)$: $f(g(x)) = \sqrt{ \frac{1}{x} - 8 } + 8$

So: $(f \circ g)(x) = \sqrt{ \frac{1}{x} - 8 } + 8$

2. $(g \circ f)(x) = g(f(x))$

This means we need to substitute $f(x)$ into $g(x)$.

From the definition of $g(x)$, we have: $g(x) = \frac{1}{x} - 8$

Now, substitute $f(x) = \sqrt{x} + 8$ into $g(x)$: $g(f(x)) = \frac{1}{\sqrt{x} + 8} - 8$

So: $(g \circ f)(x) = \frac{1}{\sqrt{x} + 8} - 8$

Final Answers:

• $(f \circ g)(x) = \sqrt{ \frac{1}{x} - 8 } + 8$
• $(g \circ f)(x) = \frac{1}{\sqrt{x} + 8} - 8$

Let me know if you need further details or clarifications on these composite functions!

Related Questions:

1. How do I determine the domain of a composite function?
2. How can I find the inverse of a composite function?
3. What are the differences between $f(x) \circ g(x)$ and $g(x) \circ f(x)$?
4. How can I graph the composite functions $f \circ g$ and $g \circ f$?
5. Can I simplify the expressions for $(f \circ g)(x)$ and $(g \circ f)(x)$?

Tip: When working with composite functions, always remember to check the domains of the individual functions and how they interact to affect the overall domain of the composite function.