We are asked to find the composite functions $(f \circ g)(x)$ and $(g \circ f)(x)$ for the given pair of functions:

• $f(x) = x + \frac{13}{2}$
• $g(x) = \sqrt{4 - x}$

1. Finding $(f \circ g)(x)$:

This represents $f(g(x))$, which means we substitute $g(x)$ into $f(x)$.

• First, recall $g(x) = \sqrt{4 - x}$.
• Now substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(\sqrt{4 - x}) = \sqrt{4 - x} + \frac{13}{2}$

Thus,

$(f \circ g)(x) = \sqrt{4 - x} + \frac{13}{2}$

2. Finding $(g \circ f)(x)$:

This represents $g(f(x))$, which means we substitute $f(x)$ into $g(x)$.

• First, recall $f(x) = x + \frac{13}{2}$.
• Now substitute $f(x)$ into $g(x)$:

$g(f(x)) = g\left(x + \frac{13}{2}\right) = \sqrt{4 - \left(x + \frac{13}{2}\right)}$

Simplify the expression inside the square root:

$g(f(x)) = \sqrt{4 - x - \frac{13}{2}} = \sqrt{\frac{8}{2} - x - \frac{13}{2}} = \sqrt{\frac{-5}{2} - x}$

Thus,

$(g \circ f)(x) = \sqrt{\frac{-5}{2} - x}$

Summary:

• $(f \circ g)(x) = \sqrt{4 - x} + \frac{13}{2}$
• $(g \circ f)(x) = \sqrt{\frac{-5}{2} - x}$

Do you need further details or have any questions?

Here are 5 related questions to explore:

1. What is the domain of $f(x)$ and $g(x)$?
2. How do you find the domain of composite functions?
3. What happens if the result inside the square root is negative?
4. Can you combine the results of $f(x)$ and $g(x)$ in a different way?
5. How does composition change if $g(x)$ is more complex?

Tip: Always check the domains when working with composite functions to avoid invalid inputs like negative square roots.