To graph the function $h(x)$ based on its piecewise definition, let's analyze each segment:

Function Definition

The function $h(x)$ is defined in five intervals:

1. $h(x) = -3$ for $-2.5 < x \leq -1.5$
2. $h(x) = -2$ for $-1.5 < x \leq -0.5$
3. $h(x) = -1$ for $-0.5 < x < 0.5$
4. $h(x) = 0$ for $0.5 \leq x < 1.5$
5. $h(x) = 1$ for $1.5 \leq x < 2.5$

Steps to Graph

1. For each interval, plot a horizontal line segment at the specified $y$-value (e.g., -3, -2, -1, etc.).
2. Indicate open and closed endpoints as specified in each interval.

Plotting Points and Line Segments

• Segment 1: From $x = -2.5$ to $x = -1.5$ at $y = -3$
  • Open circle at $(-2.5, -3)$ and closed circle at $(-1.5, -3)$.
• Segment 2: From $x = -1.5$ to $x = -0.5$ at $y = -2$
  • Open circle at $(-1.5, -2)$ and closed circle at $(-0.5, -2)$.
• Segment 3: From $x = -0.5$ to $x = 0.5$ at $y = -1$
  • Open circle at $(-0.5, -1)$ and open circle at $(0.5, -1)$.
• Segment 4: From $x = 0.5$ to $x = 1.5$ at $y = 0$
  • Closed circle at $(0.5, 0)$ and open circle at $(1.5, 0)$.
• Segment 5: From $x = 1.5$ to $x = 2.5$ at $y = 1$
  • Closed circle at $(1.5, 1)$ and open circle at $(2.5, 1)$.

Make sure to mark each endpoint correctly (open or closed) to match the domain restrictions.

Would you like further details on interpreting piecewise functions?

Further Questions

1. How do we identify open and closed intervals on a piecewise graph?
2. What is the significance of open and closed circles in graphing piecewise functions?
3. How do changes in intervals affect the continuity of a piecewise function?
4. Can a piecewise function be continuous across all intervals?
5. How do we determine the range of a piecewise function from its graph?

Tip

When graphing piecewise functions, pay careful attention to the inclusion or exclusion of endpoints in each interval, as it influences the appearance and interpretation of the function’s graph.