https://file.aiquickdraw.com/mathbot/file/1723460509415nxze5xiq.jpg

Graph this function
G(x)        Using these ( -3,0,2,3,4,6) Based the picture

To graph the function \( g(x) \) using the given values, let's first understand the piecewise function from the image:

\[
g(x) = \begin{cases} 
\frac{5}{3}x + 3 & \text{if } x \leq 0 \\
-x & \text{if } x > 0 
\end{cases}
\]

Given values: \( x = -3, 0, 2, 3, 4, 6 \).

### Step-by-Step Calculation:

1. **For \( x = -3 \) (since \( x \leq 0 \)):**
   \[
   g(-3) = \frac{5}{3}(-3) + 3 = -5 + 3 = -2
   \]

2. **For \( x = 0 \) (since \( x \leq 0 \)):**
   \[
   g(0) = \frac{5}{3}(0) + 3 = 0 + 3 = 3
   \]

3. **For \( x = 2 \) (since \( x > 0 \)):**
   \[
   g(2) = -2 = -2
   \]

4. **For \( x = 3 \) (since \( x > 0 \)):**
   \[
   g(3) = -3 = -3
   \]

5. **For \( x = 4 \) (since \( x > 0 \)):**
   \[
   g(4) = -4 = -4
   \]

6. **For \( x = 6 \) (since \( x > 0 \)):**
   \[
   g(6) = -6 = -6
   \]

### Summary of Points:
- \( g(-3) = -2 \)
- \( g(0) = 3 \)
- \( g(2) = -2 \)
- \( g(3) = -3 \)
- \( g(4) = -4 \)
- \( g(6) = -6 \)

### Graphing:

You can plot the points \((-3, -2)\), \((0, 3)\), \((2, -2)\), \((3, -3)\), \((4, -4)\), and \((6, -6)\) on a coordinate plane and draw the corresponding line segments based on the piecewise conditions.

Would you like a visual representation of this graph or further explanations?

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Here are some related questions you might find interesting:

1. What happens to the function as \( x \) approaches 0 from the left and right?
2. How would the graph change if the piecewise function was \( g(x) = \frac{5}{3}x + 3 \) for all \( x \)?
3. What is the significance of the slope in each piece of the function?
4. How does the function behave as \( x \) becomes increasingly negative or positive?
5. Can you describe the symmetry of the function, if any?
6. How would adding a constant to \( g(x) \) in each piece affect the graph?
7. What is the domain and range of this piecewise function?
8. Can the function \( g(x) \) be continuous? Why or why not?

**Tip:** When dealing with piecewise functions, carefully note the conditions for each piece to ensure you're using the correct formula for your calculations.

Learn how to graph the piecewise function g(x) using the points -3, 0, 2, 3, 4, and 6. Detailed step-by-step calculation provided for each point. Suitable for high school students.

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