Determine whether the graph below is that of a function by using the​ vertical-line test.  If it​ is, use the graph to find 
​(a)
its domain and range.
​(b)
the​ intercepts, if any.
​(c)
any symmetry with respect to the​ x-axis, y-axis, or the origin.
.
.
.
Question content area top right
Part 1
pinegative pi divided by 2pi divided by 2negative pi
1
-1



A coordinate system has a horizontal axis labeled from negative StartFraction 3 pi Over 2 EndFraction to StartFraction 3 pi Over 2 EndFraction in increments of StartFraction pi Over 2 EndFraction and a vertical axis labeled from negative 1.5 to 1.5 in increments of 1. Upper A curve starts at the point left parenthesis negative pi comma 0 right parenthesis comma rises until it reaches the point left parenthesis negative StartFraction pi Over 2 EndFraction comma 1 right parenthesis comma then falls comma passing through the origin comma until it reaches the point left parenthesis StartFraction pi Over 2 EndFraction comma negative 1 right parenthesis comma then rises and ends at the point left parenthesis pi comma 0 right parenthesis .
Question content area bottom
Part 1
Is the graph that of a​ function?
Yes
No

### 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.

Determine whether the graph below is that of a function by using the vertical-line test. If it is, use the graph to find (a) its domain and range. (b) the intercepts, if any. (c) any symmetry with respect to the x-axis, y-axis, or the origin.

Explore how to determine if a graph represents a function using the vertical-line test. Learn to find the domain, range, intercepts, and analyze symmetry with respect to the x-axis, y-axis, and origin. This comprehensive guide is suitable for high school students in Grades 10-12.

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