https://file.aiquickdraw.com/mathbot/file/1726862475770vnnrx9zo.jpg

The graphs in the image show the functions \( f(x) \) and \( g(x) \) as piecewise linear functions.

### \( f(x) \) (left graph):
- The graph is blue and shows a line segment that starts at the origin \((0,0)\) and extends to the point \((2,1)\).
- This line has a positive slope, suggesting that the function \( f(x) \) is increasing.

### \( g(x) \) (right graph):
- The graph is red and shows a line segment that also starts at the origin \((0,0)\) but decreases to the point \((2, -0.5)\).
- This indicates a negative slope, meaning the function \( g(x) \) is decreasing.

From visual inspection, the slopes of the lines can be approximated.

### Calculating the slopes:
1. For \( f(x) \):
   - Slope \( m = \frac{\text{change in } y}{\text{change in } x} = \frac{1 - 0}{2 - 0} = \frac{1}{2} \).
   - So, \( f(x) = \frac{1}{2}x \).

2. For \( g(x) \):
   - Slope \( m = \frac{-0.5 - 0}{2 - 0} = \frac{-0.5}{2} = -\frac{1}{4} \).
   - So, \( g(x) = -\frac{1}{4}x \).

Thus, the equations of the lines are approximately:
- \( f(x) = \frac{1}{2}x \)
- \( g(x) = -\frac{1}{4}x \)

Let me know if you need more details or have any questions! 

Here are five related questions to expand on this:

1. What is the significance of the slope in each function?
2. How do we determine whether a function is increasing or decreasing from its graph?
3. Can we derive the equations of the lines using other methods besides visual inspection?
4. What is the difference between a positive and negative slope in terms of the function's behavior?
5. How would the graph change if \( f(x) \) or \( g(x) \) had different slopes?

**Tip**: When analyzing a graph, always check the slope of the line, as it determines whether the function is increasing or decreasing.

The figure below to the left is a graph of f(x), and below to the right is g(x).

This problem involves interpreting and solving the linear functions f(x) and g(x) based on their graphs. Learn how to calculate slopes, understand increasing and decreasing functions, and derive the equations of linear functions from their graphs. Suitable for students in Grades 9-12.

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