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We are given a table showing values of \(x\) and their corresponding values of \(g(x)\), where \(g(x) = \frac{f(x)}{x+3}\), and we know that \(f(x)\) is a linear function. The question asks us to find the y-intercept of the graph of \(y = f(x)\), which occurs when \(x = 0\).

### Step-by-Step Solution:

#### Step 1: Analyze the table
From the table, we have the following values of \(g(x)\):

\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-27 & 3 \\
-9 & 0 \\
21 & 5 \\
\hline
\end{array}
\]

#### Step 2: Express \(f(x)\) in terms of \(g(x)\)
We know that \(g(x) = \frac{f(x)}{x + 3}\). This means:
\[
f(x) = g(x) \cdot (x + 3)
\]

Now, we'll use this relationship to find \(f(x)\) for each of the given values of \(x\).

#### Step 3: Calculate \(f(x)\) at the given points

1. For \(x = -27\):
   \[
   g(-27) = 3 \quad \text{so} \quad f(-27) = 3 \cdot (-27 + 3) = 3 \cdot (-24) = -72
   \]

2. For \(x = -9\):
   \[
   g(-9) = 0 \quad \text{so} \quad f(-9) = 0 \cdot (-9 + 3) = 0
   \]

3. For \(x = 21\):
   \[
   g(21) = 5 \quad \text{so} \quad f(21) = 5 \cdot (21 + 3) = 5 \cdot 24 = 120
   \]

Now we have the points \((-27, -72)\), \((-9, 0)\), and \((21, 120)\) for the linear function \(f(x)\).

#### Step 4: Find the equation of the linear function \(f(x)\)
The general form of a linear equation is:
\[
f(x) = mx + b
\]
where \(m\) is the slope, and \(b\) is the y-intercept. We can find the slope \(m\) using the two points \((-9, 0)\) and \((21, 120)\):
\[
m = \frac{120 - 0}{21 - (-9)} = \frac{120}{30} = 4
\]
So the equation becomes:
\[
f(x) = 4x + b
\]

#### Step 5: Solve for the y-intercept \(b\)
We can substitute one of the points, say \((-9, 0)\), into the equation \(f(x) = 4x + b\) to solve for \(b\):
\[
0 = 4(-9) + b \quad \Rightarrow \quad 0 = -36 + b \quad \Rightarrow \quad b = 36
\]

#### Step 6: Conclusion
The y-intercept of the graph of \(y = f(x)\) is \(b = 36\), so the correct answer is:

\[
\boxed{A: (0, 36)}
\]

---

Would you like further details on any part of the solution? Here are five related questions to expand your understanding:

1. How do you find the equation of a line given two points?
2. What is the significance of the slope in a linear equation?
3. Can the method used here be applied to non-linear functions?
4. How do transformations of functions affect the intercepts?
5. What is the difference between the x-intercept and y-intercept in a graph?

**Tip:** Always double-check the function given in a question. Often, understanding the relationship between different variables is key to solving a problem efficiently.

The table shows three values of x and their corresponding values of g(x), where g(x) = f(x) / (x + 3), and f is a linear function. What is the y-intercept of the graph of y = f(x) in the xy-plane?

Learn how to find the y-intercept of a linear function given a table of values for g(x) = f(x) / (x + 3). This problem covers key algebraic concepts and solving linear functions, suitable for high school math students in Grades 10-12.

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