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 $(0, 40)$.

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.