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Let's go through each problem step-by-step to find the domain, range, and determine whether each relation is a function.

### Problem 1
- **Relation**: 
  - \(x\) values: \(-5, -3, 0, 4\)
  - \(y\) values: \(-2, -2, 1, 7\)
- **Domain (D)**: The set of all \(x\) values: \(\{-5, -3, 0, 4\}\)
- **Range (R)**: The set of all \(y\) values: \(\{-2, 1, 7\}\)
- **Function?**: **No**, because the \(x\) value \(-5\) is mapped to more than one \(y\) value (\(-2\) and \(1\)).

### Problem 2
- **Graph**: A set of points on a coordinate grid.
  - Points: \((-2, 3), (0, 1), (1, -3), (3, 4), (5, -1)\)
- **Domain (D)**: The set of all \(x\) values: \(\{-2, 0, 1, 3, 5\}\)
- **Range (R)**: The set of all \(y\) values: \(\{-3, -1, 1, 3, 4\}\)
- **Function?**: **Yes**, because each \(x\) value corresponds to exactly one \(y\) value.

### Problem 3
- **Set of Points**: \(\{(-3, -9), (-2, -1), (-2, -6), (-1, -9)\}\)
- **Domain (D)**: The set of all \(x\) values: \(\{-3, -2, -1\}\)
- **Range (R)**: The set of all \(y\) values: \(\{-9, -6, -1\}\)
- **Function?**: **No**, because the \(x\) value \(-2\) is mapped to two different \(y\) values (\(-1\) and \(-6\)).

### Problem 4
- **Graph**: A downward-opening parabola.
- **Domain (D)**: All real numbers, \((-\infty, \infty)\)
- **Range (R)**: All \(y \leq 2\) (the parabola reaches its maximum at \(y = 2\)), so the range is \((-\infty, 2]\).
- **Function?**: **Yes**, because for every \(x\) value, there is exactly one \(y\) value.

### Problem 5
- **Graph**: A line with a positive slope.
- **Domain (D)**: All real numbers, \((-\infty, \infty)\)
- **Range (R)**: All real numbers, \((-\infty, \infty)\)
- **Function?**: **Yes**, because a line (except for a vertical line) represents a function.

### Problem 6
- **Graph**: A vertical parabola shifted to the left.
- **Domain (D)**: All real numbers, \((-\infty, \infty)\)
- **Range (R)**: All real numbers, \((-\infty, \infty)\)
- **Function?**: **No**, because there are some \(x\) values (like near the vertex) that map to more than one \(y\) value.

### Problem 7
- **Graph**: A piecewise linear function with a horizontal line segment.
- **Domain (D)**: From the leftmost point to the rightmost point: \([-4, 4]\)
- **Range (R)**: From the lowest point to the highest point: \([0, 4]\)
- **Function?**: **No**, because there is a vertical line segment that touches the graph at more than one point.

### Problem 8
- **Graph**: A "V"-shaped graph (absolute value function).
- **Domain (D)**: All real numbers, \((-\infty, \infty)\)
- **Range (R)**: All \(y \geq -2\), so the range is \([-2, \infty)\).
- **Function?**: **Yes**, because each \(x\) value corresponds to exactly one \(y\) value.

### Problem 9
- **Graph**: A function with an open circle (undefined point) and two separate curves.
- **Domain (D)**: All real numbers except where the hole is located, e.g., \((-\infty, 2) \cup (2, \infty)\) if the hole is at \(x = 2\).
- **Range (R)**: All real numbers except for the \(y\)-value at the hole point.
- **Function?**: **Yes**, assuming the open circle represents a point where the function is not defined.

Would you like more details or have any specific questions?

### Related Questions:
1. How do you determine if a relation is a function from a graph?
2. What are the key characteristics of functions in terms of their domain and range?
3. How does the vertical line test work to determine if a graph represents a function?
4. How can you write the domain and range in interval notation?
5. Why are some relations not considered functions?

### Tip:
Always remember the "Vertical Line Test" — if any vertical line intersects a graph more than once, the graph does not represent a function.

Give the domain and range of each relation and determine whether it is a function for the provided graphs and sets of points.

Learn how to determine the domain, range, and whether a relation is a function using graphs and sets of points. Includes step-by-step solutions for problems involving set and interval notation, with an emphasis on the vertical line test.

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