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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**.

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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.

Jika hasil refleksi dari fungsi linear f(x) = -1/2x + 8 terhadap garis y = -x adalah g(x), tentukan nilai g(4)!

Solve for g(4) by finding the reflection of the linear function f(x) = -1/2x + 8 over the line y = -x. This solution includes finding the inverse function and calculating g(x) = f^{-1}(-x). Suitable for students in Grades 10-12.

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