To determine the piecewise function from this graph, let's analyze its structure step-by-step.

1. Segment Analysis:

   • The graph consists of three distinct segments.
   • The first segment starts at $(-1, -1)$ and moves linearly to $(1, 1)$.
   • The second segment is a horizontal line that extends from $(1, 1)$ to $(5, 1)$.
   • The third segment is a single point at $(5, 0)$, where the function seems to "reset."

2. Defining Each Segment:

   • For $x$ in $[-1, 1]$, the line segment appears to have a slope of $1$, as it rises 2 units over a 2-unit span in $x$.
     • Equation for the first segment: $f(x) = x$ for $[-1 \leq x \leq 1]$.
   • For $x$ in $(1, 5)$, the function is constant.
     • Equation for the second segment: $f(x) = 1$ for $1 < x < 5$.
   • At $x = 5$, the function value drops to zero, suggesting a discrete point.
     • Equation for the third segment: $f(x) = 0$ at $x = 5$.

3. Piecewise Function: Combining these observations, the piecewise function is:

   x & \text{for } -1 \leq x \leq 1, \\ 1 & \text{for } 1 < x < 5, \\ 0 & \text{for } x = 5. \end{cases}$$

Would you like further details or have any questions about this interpretation?

Here are some additional questions for you:

1. How would the function change if the slope in the first segment were steeper?
2. What effect would shifting the horizontal segment vertically have on the overall function?
3. How can you determine the slope of a line segment from a graph?
4. What if the third segment continued horizontally instead of resetting to zero at $x = 5$?
5. How can we represent open and closed intervals in piecewise functions?

Tip: When analyzing piecewise functions, always check each segment's domain and continuity at the endpoints for any jumps or discontinuities.