The problem asks for the reflection of the linear function $f(x) = -\frac{1}{2}x + 8$ with respect to the line $y = -x$, and it requires determining $g(4)$, where $g(x)$ is the reflected function.

Step-by-step Solution:

1. Find the reflection rules: When a function is reflected over the line $y = -x$, the coordinates $(x, y)$ transform as follows:

   • The new coordinates after reflection are $(y, x)$. In terms of functions, this means the reflection of $f(x)$ across the line $y = -x$ can be written as: $g(x) = f^{-1}(-x)$
   • So, to find the reflection, we need to first find the inverse of $f(x)$.

2. Find the inverse of $f(x)$:
   The given function is $f(x) = -\frac{1}{2}x + 8$. To find the inverse:

   • Set $y = -\frac{1}{2}x + 8$.
   • Solve for $x$: $y = -\frac{1}{2}x + 8$ $y - 8 = -\frac{1}{2}x$ $x = -2(y - 8) = -2y + 16$ Therefore, the inverse function is: $f^{-1}(x) = -2x + 16$

3. Find $g(x) = f^{-1}(-x)$:
   Substitute $-x$ into the inverse function: $g(x) = f^{-1}(-x) = -2(-x) + 16 = 2x + 16$

4. Calculate $g(4)$: Now, substitute $x = 4$ into $g(x)$: $g(4) = 2(4) + 16 = 8 + 16 = 24$

Thus, the value of $g(4)$ is 24.

Would you like more details on any step or have further questions?

Here are five related questions for you to explore:

1. How do you find the reflection of a linear function over other lines like $y = x$?
2. What is the geometric meaning of reflecting a function over the line $y = -x$?
3. How do you find the inverse of a more complex function, such as a quadratic?
4. What are the properties of functions and their inverses in terms of domain and range?
5. Can the reflection of a non-linear function, such as $f(x) = x^2$, also be calculated similarly?

Tip: Always ensure that the function is one-to-one before finding its inverse. If a function is not one-to-one, you may need to restrict its domain.