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We are given the piecewise function \( g(x) \):

\[
g(x) = 
\begin{cases} 
3 - x & \text{for } x < 3 \\
4x - 12 & \text{for } x \geq 3
\end{cases}
\]

We need to determine \( g'(3) \), which means checking whether the function is differentiable at \( x = 3 \).

### Step 1: Check continuity at \( x = 3 \)
First, for the function to be differentiable at \( x = 3 \), it must be continuous there. Let's check the left-hand and right-hand limits of \( g(x) \) as \( x \to 3 \).

- Left-hand limit as \( x \to 3^- \): 
  Using \( g(x) = 3 - x \) for \( x < 3 \),
  \[
  \lim_{x \to 3^-} g(x) = 3 - 3 = 0
  \]

- Right-hand limit as \( x \to 3^+ \): 
  Using \( g(x) = 4x - 12 \) for \( x \geq 3 \),
  \[
  \lim_{x \to 3^+} g(x) = 4(3) - 12 = 12 - 12 = 0
  \]

Since both the left-hand and right-hand limits are equal, the function is continuous at \( x = 3 \).

### Step 2: Check differentiability at \( x = 3 \)
Now, let's check if the derivative exists at \( x = 3 \). We'll compute the left-hand and right-hand derivatives.

- Left-hand derivative at \( x = 3 \): 
  The derivative of \( 3 - x \) is \( -1 \), so
  \[
  \lim_{x \to 3^-} g'(x) = -1
  \]

- Right-hand derivative at \( x = 3 \): 
  The derivative of \( 4x - 12 \) is \( 4 \), so
  \[
  \lim_{x \to 3^+} g'(x) = 4
  \]

Since the left-hand and right-hand derivatives are not equal, the derivative does not exist at \( x = 3 \).

### Conclusion:
The correct answer is that \( g'(3) \) is **nonexistent**, corresponding to option **D**.

---

Would you like further details or have any questions? Here are 5 related questions to expand on this problem:

1. What conditions must be met for a piecewise function to be differentiable at a point?
2. How can you determine if a piecewise function is continuous at a point?
3. Why is the left-hand and right-hand derivative comparison crucial for differentiability?
4. Can a function be continuous but not differentiable at the same point? If so, provide an example.
5. How would the derivative behave if the piecewise components had the same slope at the boundary?

**Tip**: To determine differentiability, always check both continuity and the equality of left-hand and right-hand derivatives at the point in question.

If g is the function defined above, then g'(3) is?

This problem explores the differentiability of a piecewise function g(x) at x = 3. It involves checking the continuity and comparing the left-hand and right-hand derivatives. Learn how to approach such problems step by step.

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