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.