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To solve the problem in the image, we need to focus on determining the **y-intercept** of the function \( y = f(x) \) given the relationship \( g(x) = \frac{f(x)}{x + 5} \), and the table with corresponding values of \(x\) and \( g(x) \).

### Step-by-step breakdown:

1. **Identify the relationship between \( f(x) \) and \( g(x) \):**
   \[
   g(x) = \frac{f(x)}{x + 5}
   \]
   From this, we can rearrange the equation to express \( f(x) \) in terms of \( g(x) \) and \( x \):
   \[
   f(x) = g(x)(x + 5)
   \]

2. **Use the values from the table:**
   Let's compute \( f(x) \) for each \( x \) and \( g(x) \) value from the table:
   - For \( x = -20 \), \( g(-20) = 4 \):
     \[
     f(-20) = 4 \times (-20 + 5) = 4 \times (-15) = -60
     \]
   - For \( x = -8 \), \( g(-8) = 0 \):
     \[
     f(-8) = 0 \times (-8 + 5) = 0
     \]
   - For \( x = 10 \), \( g(10) = 6 \):
     \[
     f(10) = 6 \times (10 + 5) = 6 \times 15 = 90
     \]

   So, the points we have for \( f(x) \) are:
   - \( (-20, -60) \)
   - \( (-8, 0) \)
   - \( (10, 90) \)

3. **Find the equation of the linear function \( f(x) \):**
   We now have two points that we can use to find the slope \( m \). Using the points \( (-8, 0) \) and \( (10, 90) \), the slope is:
   \[
   m = \frac{90 - 0}{10 - (-8)} = \frac{90}{18} = 5
   \]

   Now that we have the slope \( m = 5 \), we can use the point-slope form of the equation \( f(x) = mx + b \), where \( b \) is the y-intercept. Using the point \( (-8, 0) \):
   \[
   0 = 5(-8) + b
   \]
   \[
   0 = -40 + b
   \]
   \[
   b = 40
   \]

   Thus, the equation of the line is:
   \[
   f(x) = 5x + 40
   \]

4. **Determine the y-intercept:**
   The y-intercept is the value of \( f(x) \) when \( x = 0 \):
   \[
   f(0) = 5(0) + 40 = 40
   \]

So, the **y-intercept** of the graph of \( y = f(x) \) is \( (0, 40) \).

### Final Answer:
The correct choice is **D** \( (0, 40) \).

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Would you like further details on any part of this solution?

Here are 5 questions to expand your understanding:
1. How would the solution change if \( f(x) \) were quadratic instead of linear?
2. Can we derive the slope of a line using any two points on the line? Why?
3. What is the significance of the y-intercept in a linear function?
4. How can we use the point-slope form of a line to solve other types of problems?
5. What happens to the slope of a line as the line becomes more vertical?

**Tip:** Whenever you are given a linear function, the y-intercept is simply the value of \( y \) when \( x = 0 \). Always solve for that first to save time.

The table shows three values of x and their corresponding values of g(x), where g(x) = f(x) / (x + 5) and f is a linear function. What is the y-intercept of the graph of y = f(x) in the xy-plane?

Solve for the y-intercept of the function y = f(x) using given values of x and g(x) in the table, where g(x) = f(x) / (x + 5). Learn to determine the linear function f(x), calculate its slope, and identify the intercept. This algebra problem is suitable for students in grades 9-12.

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