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We are given a piecewise function \( g(x) \) and need to find the values of \( g(-4) \), \( g(3) \), and \( g(6) \). The function is defined as:

\[
g(x) = \begin{cases} 
x + 4 & \text{for } x \leq 3 \\
9 - x & \text{for } x > 3
\end{cases}
\]

### Step 1: Find \( g(-4) \)
Since \( -4 \leq 3 \), we use the first case of the piecewise function:
\[
g(-4) = -4 + 4 = 0
\]

### Step 2: Find \( g(3) \)
Since \( 3 \leq 3 \), we again use the first case:
\[
g(3) = 3 + 4 = 7
\]

### Step 3: Find \( g(6) \)
Since \( 6 > 3 \), we use the second case of the piecewise function:
\[
g(6) = 9 - 6 = 3
\]

### Final Values:
- \( g(-4) = 0 \)
- \( g(3) = 7 \)
- \( g(6) = 3 \)

Would you like further clarification on any part of this process?

Here are five related questions to deepen your understanding:
1. What happens if we define a new piecewise function for \( x > 6 \)?
2. How would the function behave if we modified the conditions, for example, using \( x + 2 \) instead of \( x + 4 \)?
3. Can you graph the piecewise function for \( x = -4 \) to \( x = 6 \)?
4. How does the function change if we introduce another condition at \( x = 0 \)?
5. What is the domain and range of this piecewise function?

**Tip:** For piecewise functions, always check the condition associated with the input value carefully to select the correct function definition.

For the piecewise function, find the values g(-4), g(3), and g(6). g(x) is defined as: g(x) = {x + 4, for x ≤ 3; 9 - x, for x > 3}.

Learn how to evaluate a piecewise function for specific values such as g(-4), g(3), and g(6). This algebra problem involves understanding and applying piecewise function rules for different ranges of x. Suitable for students in Grades 8-10.

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