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.