We are given the functions $f(x) = \sqrt{x} - 8$ and $g(x) = x^2 + 4$, and we need to compute the following compositions:

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

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

This means we substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(x^2 + 4)$

Now, replace $x$ in $f(x) = \sqrt{x} - 8$ with $x^2 + 4$:

$f(x^2 + 4) = \sqrt{x^2 + 4} - 8$

Thus:

$(f \circ g)(x) = \sqrt{x^2 + 4} - 8$

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

This means we substitute $f(x)$ into $g(x)$:

$g(f(x)) = g(\sqrt{x} - 8)$

Now, replace $x$ in $g(x) = x^2 + 4$ with $\sqrt{x} - 8$:

$g(\sqrt{x} - 8) = (\sqrt{x} - 8)^2 + 4$

First, expand $(\sqrt{x} - 8)^2$:

$(\sqrt{x} - 8)^2 = (\sqrt{x})^2 - 2 \cdot \sqrt{x} \cdot 8 + 8^2 = x - 16\sqrt{x} + 64$

Now, add 4:

$g(\sqrt{x} - 8) = x - 16\sqrt{x} + 64 + 4 = x - 16\sqrt{x} + 68$

Thus:

$(g \circ f)(x) = x - 16\sqrt{x} + 68$

Final Answers:

1. $(f \circ g)(x) = \sqrt{x^2 + 4} - 8$
2. $(g \circ f)(x) = x - 16\sqrt{x} + 68$

Would you like more details on any step or further explanations?

Follow-up Questions:

1. What is the general definition of a function composition?
2. Can all functions be composed, or are there restrictions?
3. How do the domains of the functions affect the composition of functions?
4. What would the graph of $f \circ g$ look like?
5. How can you simplify or analyze composite functions using algebraic techniques?

Tip: When composing functions, always check the domain of the resulting function, as it might be more restricted than the individual domains of the functions involved.