Practice Worksheet: Relations & Functions - Exercises to convert between different representations and determine function status.

Problem Analysis:

The worksheet focuses on understanding relations and functions. A relation is any set of ordered pairs, while a function is a special type of relation where each input (x-value) corresponds to exactly one output (y-value). The task involves converting between different representations of relations (set of ordered pairs, table, graph, mapping diagram) and determining whether the relation is a function.

Additionally, for graphs, we need to identify the domain, range, and determine if the graph represents a function using the vertical line test:
- If a vertical line can intersect the graph at more than one point, it is not a function.
- If every vertical line intersects the graph at most once, it is a function.

---

Solution:

#### Part 1: Completing the Forms and Determining if the Relation is a Function

##### Problem 1:
- Set of ordered pairs: \(\{(-2, -1), (2, 1), (-1, -2), (1, 2)\}\)
- Table:
$$
\begin{array}{c|c}
x & y \\
\hline
-2 & -1 \\
2 & 1 \\
-1 & -2 \\
1 & 2 \\
\end{array}
$$
- Graph: Plot the points \((-2, -1)\), \((2, 1)\), \((-1, -2)\), and \((1, 2)\) on the coordinate plane.
- Mapping Diagram: Draw two ovals, one for \(x\)-values and one for \(y\)-values. Connect each \(x\) to its corresponding \(y\):
- \(-2 \rightarrow -1\)
- \(2 \rightarrow 1\)
- \(-1 \rightarrow -2\)
- \(1 \rightarrow 2\)
- Function?: Yes, because each \(x\)-value maps to exactly one \(y\)-value.

##### Problem 2:
- Table:
$$
\begin{array}{c|c}
x & y \\
\hline
1 & -2 \\
-3 & -1 \\
1 & 0 \\
2 & 2 \\
0 & 3 \\
\end{array}
$$
- Set of ordered pairs: \(\{(1, -2), (-3, -1), (1, 0), (2, 2), (0, 3)\}\)
- Graph: Plot the points \((1, -2)\), \((-3, -1)\), \((1, 0)\), \((2, 2)\), and \((0, 3)\).
- Mapping Diagram: Draw two ovals, one for \(x\)-values and one for \(y\)-values. Connect each \(x\) to its corresponding \(y\):
- \(1 \rightarrow -2\)
- \(-3 \rightarrow -1\)
- \(1 \rightarrow 0\)
- \(2 \rightarrow 2\)
- \(0 \rightarrow 3\)
- Function?: No, because the \(x\)-value \(1\) maps to two different \(y\)-values (\(-2\) and \(0\)).

##### Problem 3:
- Graph: The graph shows points at \((-2, -1)\), \((-1, 0)\), \((0, 1)\), \((1, 2)\), and \((2, 3)\).
- Set of ordered pairs: \(\{(-2, -1), (-1, 0), (0, 1), (1, 2), (2, 3)\}\)
- Table:
$$
\begin{array}{c|c}
x & y \\
\hline
-2 & -1 \\
-1 & 0 \\
0 & 1 \\
1 & 2 \\
2 & 3 \\
\end{array}
$$
- Mapping Diagram: Draw two ovals, one for \(x\)-values and one for \(y\)-values. Connect each \(x\) to its corresponding \(y\):
- \(-2 \rightarrow -1\)
- \(-1 \rightarrow 0\)
- \(0 \rightarrow 1\)
- \(1 \rightarrow 2\)
- \(2 \rightarrow 3\)
- Function?: Yes, because each \(x\)-value maps to exactly one \(y\)-value.

##### Problem 4:
- Mapping Diagram: The diagram shows:
- \(-1 \rightarrow -2\)
- \(1 \rightarrow -1\)
- \(2 \rightarrow 0\)
- \(3 \rightarrow 2\)
- Set of ordered pairs: \(\{(-1, -2), (1, -1), (2, 0), (3, 2)\}\)
- Table:
$$
\begin{array}{c|c}
x & y \\
\hline
-1 & -2 \\
1 & -1 \\
2 & 0 \\
3 & 2 \\
\end{array}
$$
- Graph: Plot the points \((-1, -2)\), \((1, -1)\), \((2, 0)\), and \((3, 2)\).
- Function?: Yes, because each \(x\)-value maps to exactly one \(y\)-value.

---

#### Part 2: Analyzing Graphs

##### Problem 5:
- Graph: A straight line with points \((-3, -3)\), \((0, 0)\), and \((2, 2)\).
- Domain: All real numbers (\(x \in \mathbb{R}\)).
- Range: All real numbers (\(y \in \mathbb{R}\)).
- Function?: Yes, because it passes the vertical line test.

##### Problem 6:
- Graph: A circle centered at the origin with radius 1.
- Domain: \([-1, 1]\)
- Range: \([-1, 1]\)
- Function?: No, because a vertical line can intersect the circle at more than one point (e.g., \(x = 0\) intersects at \((0, 1)\) and \((0, -1)\)).

##### Problem 7:
- Graph: A parabola opening upwards with vertex at \((0, -2)\) and endpoints at \((-2, 0)\) and \((2, 0)\).
- Domain: \([-2, 2]\)
- Range: \([-2, 0]\)
- Function?: Yes, because it passes the vertical line test.

---

Final Answers:

1.
- Table:
$$
\begin{array}{c|c}
x & y \\
\hline
-2 & -1 \\
2 & 1 \\
-1 & -2 \\
1 & 2 \\
\end{array}
$$
- Graph: Plot points \((-2, -1)\), \((2, 1)\), \((-1, -2)\), \((1, 2)\).
- Mapping Diagram: Connect \(-2 \rightarrow -1\), \(2 \rightarrow 1\), \(-1 \rightarrow -2\), \(1 \rightarrow 2\).
- Function?: Yes.

2.
- Set of ordered pairs: \(\{(1, -2), (-3, -1), (1, 0), (2, 2), (0, 3)\}\)
- Graph: Plot points \((1, -2)\), \((-3, -1)\), \((1, 0)\), \((2, 2)\), \((0, 3)\).
- Mapping Diagram: Connect \(1 \rightarrow -2\), \(-3 \rightarrow -1\), \(1 \rightarrow 0\), \(2 \rightarrow 2\), \(0 \rightarrow 3\).
- Function?: No.

3.
- Set of ordered pairs: \(\{(-2, -1), (-1, 0), (0, 1), (1, 2), (2, 3)\}\)
- Table:
$$
\begin{array}{c|c}
x & y \\
\hline
-2 & -1 \\
-1 & 0 \\
0 & 1 \\
1 & 2 \\
2 & 3 \\
\end{array}
$$
- Mapping Diagram: Connect \(-2 \rightarrow -1\), \(-1 \rightarrow 0\), \(0 \rightarrow 1\), \(1 \rightarrow 2\), \(2 \rightarrow 3\).
- Function?: Yes.

4.
- Set of ordered pairs: \(\{(-1, -2), (1, -1), (2, 0), (3, 2)\}\)
- Table:
$$
\begin{array}{c|c}
x & y \\
\hline
-1 & -2 \\
1 & -1 \\
2 & 0 \\
3 & 2 \\
\end{array}
$$
- Graph: Plot points \((-1, -2)\), \((1, -1)\), \((2, 0)\), \((3, 2)\).
- Function?: Yes.

5.
- Domain: \(\mathbb{R}\)
- Range: \(\mathbb{R}\)
- Function?: Yes.

6.
- Domain: \([-1, 1]\)
- Range: \([-1, 1]\)
- Function?: No.

7.
- Domain: \([-2, 2]\)
- Range: \([-2, 0]\)
- Function?: Yes.

Final Boxed Answers:

\[
\boxed{
\begin{array}{l}
\text{1. Function: Yes} \\
\text{2. Function: No} \\
\text{3. Function: Yes} \\
\text{4. Function: Yes} \\
\text{5. Domain: } \mathbb{R}, \text{ Range: } \mathbb{R}, \text{ Function: Yes} \\
\text{6. Domain: } [-1, 1], \text{ Range: } [-1, 1], \text{ Function: No} \\
\text{7. Domain: } [-2, 2], \text{ Range: } [-2, 0], \text{ Function: Yes} \\
\end{array}
}
\]