Part 1: Is the graph that of a function?

To determine whether a graph represents a function, we can use the vertical-line test. This test states that if any vertical line intersects the graph at more than one point, the graph is not a function.

Based on the description of the curve:

• The graph does not have any vertical lines that intersect the curve at more than one point.

Thus, the graph is that of a function.

---

Part 2: Domain and Range

(a) Domain:
The domain consists of all possible input values (x-values) over which the function is defined. From the description, the graph starts at $x = -\pi$ and ends at $x = \pi$, so the domain is: $\boxed{[-\pi, \pi]}$

Range:
The range consists of all possible output values (y-values) the function can take. The graph reaches a maximum of 1 at $x = -\frac{\pi}{2}$ and a minimum of -1 at $x = \frac{\pi}{2}$. Thus, the range is: $\boxed{[-1, 1]}$

---

Part 3: Intercepts

(b) Intercepts:

• x-intercepts: These occur where the graph crosses the x-axis (i.e., where $y = 0$). Based on the description, the graph crosses the x-axis at: $x = -\pi, \, 0, \, \pi$ So, the x-intercepts are: $\boxed{(-\pi, 0), (0, 0), (\pi, 0)}$

• y-intercept: This occurs where the graph crosses the y-axis (i.e., where $x = 0$). From the description, the y-intercept is: $\boxed{(0, 0)}$

---

Part 4: Symmetry

(c) Symmetry:

• Symmetry with respect to the y-axis: A graph has symmetry with respect to the y-axis if it mirrors evenly about the y-axis. This function does not exhibit y-axis symmetry because the graph is not the same on both sides of the y-axis.

• Symmetry with respect to the x-axis: A graph has symmetry with respect to the x-axis if flipping it about the x-axis yields the same graph. This graph does not have x-axis symmetry either.

• Symmetry with respect to the origin: A graph has origin symmetry if rotating it 180° about the origin produces the same graph. The graph does exhibit origin symmetry, as the points reflect around the origin in opposite quadrants.

Thus, the graph has symmetry with respect to the origin.

---

Summary of answers:

1. The graph is a function.
2. (a) Domain: $[-\pi, \pi]$, Range: $[-1, 1]$
3. (b) Intercepts: $(-\pi, 0), (0, 0), (\pi, 0)$
4. (c) Symmetry: The graph has origin symmetry.

---

Would you like more details or clarifications?

---

Related Questions:

1. How do you use the vertical-line test on different types of graphs?
2. What is the significance of the domain and range in real-world problems?
3. How can we determine the symmetry of more complex functions?
4. What are the different types of function intercepts and how are they found?
5. How does symmetry affect the overall behavior of a function graph?

Tip: The vertical-line test is a quick and reliable way to check if a curve represents a function.