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The graph provided shows two functions, \( f(x) \) which is identified as \( |x| \), and \( g(x) \), which appears as a linear piecewise function. We need to determine the formula for \( g(x) \) by examining the graph:

### Observations
1. The graph of \( g(x) \) is made of two distinct linear segments.
2. The first segment, from \( x = -8 \) to \( x = 0 \), rises from \( y = -8 \) to \( y = 0 \).
3. The second segment, from \( x = 0 \) to \( x = 4 \), rises from \( y = 0 \) to \( y = 8 \).

### Determining the Slopes
1. **For the first segment**:
   - The slope \( m \) can be calculated as \( m = \frac{\Delta y}{\Delta x} \).
   - \( \Delta y = 0 - (-8) = 8 \).
   - \( \Delta x = 0 - (-8) = 8 \).
   - Thus, \( m = \frac{8}{8} = 1 \).

   So, the equation for the first segment is \( y = x + 8 \) (since when \( x = -8 \), \( y = 0 \)).

2. **For the second segment**:
   - \( \Delta y = 8 - 0 = 8 \).
   - \( \Delta x = 4 - 0 = 4 \).
   - Thus, \( m = \frac{8}{4} = 2 \).

   The equation for the second segment is \( y = 2x \) (since when \( x = 0 \), \( y = 0 \)).

### Piecewise Function
Thus, the function \( g(x) \) is a piecewise function that can be defined as:
\[
g(x) = \begin{cases} 
x + 8 & \text{for } -8 \leq x < 0 \\
2x & \text{for } 0 \leq x \leq 4 
\end{cases}
\]

This piecewise function matches the observed segments in the graph.

If you need more details or have any questions, please let me know. Here are some related questions you might consider:

1. How would you calculate the x-intercept for each segment of \( g(x) \)?
2. What is the domain and range of the function \( g(x) \)?
3. How would the function \( g(x) \) change if the line extends beyond \( x = 4 \) and \( x = -8 \)?
4. Could \( g(x) \) represent a real-world scenario? If so, what could it represent?
5. How would you graph the derivative of \( g(x) \) based on the slopes of each segment?

**Tip:** To fully understand the behavior of piecewise functions, it is beneficial to also graph these functions manually to see how they transition between different segments.

The graphs of f and g are given. Find a formula for the function g.

Learn how to determine the formula for the piecewise function g(x) based on the provided graph. This problem covers linear function properties, graph analysis, and the creation of piecewise functions, suitable for high school students (Grades 9-12).

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