Question: Find the domain and the range of the relation shown on the graph to the right. Use the vertical line test to determine whether the graph is the graph of a function.
Determine the domain of the relation. Select the correct choice below and fill in the answer box to complete your choice.
A. The domain of the relation is the interval
(Type your answer in interval notation.)
B. The domain of the relation is the single value
(Type an integer or simplified fraction.)
Graph shape: a vertical line at x = -4 with arrows up and down, so the relation is a vertical line; position_hint = center.
1. **State the problem:** We are given a graph of a vertical line at $x = -4$ and asked to find the domain and range of the relation, and to determine if it is a function using the vertical line test.
2. **Recall definitions:**
- The **domain** of a relation is the set of all possible $x$-values.
- The **range** of a relation is the set of all possible $y$-values.
- The **vertical line test** states that if any vertical line intersects the graph more than once, the graph is not a function.
3. **Analyze the graph:**
- The graph is a vertical line at $x = -4$ extending infinitely up and down.
- This means the $x$-value is always $-4$.
- The $y$-values can be any real number.
4. **Domain:**
- Since the line is vertical at $x = -4$, the domain is the single value $x = -4$.
- In interval notation, this is written as $\{ -4 \}$ or simply the single value $-4$.
5. **Range:**
- The line extends infinitely in the $y$-direction, so the range is all real numbers.
- In interval notation, the range is $(-\infty, \infty)$.
6. **Vertical line test:**
- The graph itself is a vertical line.
- Any vertical line drawn at $x = -4$ intersects the graph infinitely many times.
- Therefore, the graph **fails** the vertical line test and is **not** a function.
**Final answers:**
- Domain: $\{ -4 \}$ (single value)
- Range: $(-\infty, \infty)$
- Not a function by vertical line test.