Homework assignment on relations, domain, and range with problems involving sets, tables, mappings, and graphs.

Let's solve each problem step by step. This homework focuses on relations, domain, range, and functions, and asks you to:

1. Identify the domain and range of a relation.
2. Represent the relation as a table, mapping diagram, and graph.
3. For graphs, identify the domain and range.

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Problem 1:

Given relation:
$$
\{(1, -4), (-4, -1), (-2, 1), (0, 4), (1, 3)\}
$$

#### Step 1: Domain and Range
- Domain: All $x$-values: $ \{1, -4, -2, 0, 1\} $
- But duplicates are removed → $ \{-4, -2, 0, 1\} $
- Range: All $y$-values: $ \{-4, -1, 1, 4, 3\} $
- No duplicates → $ \{-4, -1, 1, 3, 4\} $

✔ Domain: $ \{-4, -2, 0, 1\} $
✔ Range: $ \{-4, -1, 1, 3, 4\} $

#### Step 2: Table
| x | y |
|----|----|
| 1 | -4 |
| -4 | -1 |
| -2 | 1 |
| 0 | 4 |
| 1 | 3 |

> Note: Two different $y$-values for $x = 1$, so this is not a function.

#### Step 3: Mapping Diagram
Draw two ovals:
- Left oval: Domain values: $-4, -2, 0, 1$
- Right oval: Range values: $-4, -1, 1, 3, 4$

Draw arrows:
- $1 \to -4$
- $1 \to 3$
- $-4 \to -1$
- $-2 \to 1$
- $0 \to 4$

(You can draw this on paper.)

#### Step 4: Graph
Plot each point:
- (1, -4)
- (-4, -1)
- (-2, 1)
- (0, 4)
- (1, 3)

Plot these on the coordinate plane.

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Problem 2:

Given relation:
$$
\{(-3, -4), (-1, 2), (0, 0), (-3, 5), (2, 4)\}
$$

#### Step 1: Domain and Range
- Domain: $x$-values: $-3, -1, 0, -3, 2$ → remove duplicate → $ \{-3, -1, 0, 2\} $
- Range: $y$-values: $-4, 2, 0, 5, 4$ → no duplicates → $ \{-4, 0, 2, 4, 5\} $

✔ Domain: $ \{-3, -1, 0, 2\} $
✔ Range: $ \{-4, 0, 2, 4, 5\} $

#### Step 2: Table
| x | y |
|----|----|
| -3 | -4 |
| -1 | 2 |
| 0 | 0 |
| -3 | 5 |
| 2 | 4 |

Again, $x = -3$ maps to two $y$-values → not a function

#### Step 3: Mapping Diagram
Left oval: $-3, -1, 0, 2$
Right oval: $-4, 0, 2, 4, 5$

Arrows:
- $-3 \to -4$
- $-3 \to 5$
- $-1 \to 2$
- $0 \to 0$
- $2 \to 4$

#### Step 4: Graph
Plot all points:
- (-3, -4)
- (-1, 2)
- (0, 0)
- (-3, 5)
- (2, 4)

Note: Two points at $x = -3$: one at $y = -4$, one at $y = 5$. So vertical line test fails → not a function.

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Graph Problems (3–8)

We now analyze the graphs.

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Problem 3:

Scattered points:
Points:
(-4, 1), (-2, 0), (-1, 3), (0, -1), (1, 2), (2, 3)

#### Domain: $x$-values: $ \{-4, -2, -1, 0, 1, 2\} $
#### Range: $y$-values: $ \{1, 0, 3, -1, 2\} $ → $ \{-1, 0, 1, 2, 3\} $

✔ Domain: $ \{-4, -2, -1, 0, 1, 2\} $
✔ Range: $ \{-1, 0, 1, 2, 3\} $

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Problem 4:

Points:
(-4, 1), (-3, 2), (-2, 3), (-1, 2), (0, 1)

This looks like a "V" shape, symmetric.

#### Domain: $x$-values: $ \{-4, -3, -2, -1, 0\} $
#### Range: $y$-values: $ \{1, 2, 3\} $

✔ Domain: $ \{-4, -3, -2, -1, 0\} $
✔ Range: $ \{1, 2, 3\} $

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Problem 5:

Line from (-4, -2) to (4, 4)

It's a straight line with positive slope.

#### Domain: All $x$ from -4 to 4 → $ [-4, 4] $
#### Range: All $y$ from -2 to 4 → $ [-2, 4] $

✔ Domain: $ [-4, 4] $
✔ Range: $ [-2, 4] $

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Problem 6:

V-shaped graph, vertex at origin, opens to the right.

This is a horizontal V-shape, like $x = |y|$ or similar.

Points:
(0, -3), (1, -2), (2, -1), (3, 0), (2, 1), (1, 2), (0, 3)

So it’s symmetric about $x$-axis.

#### Domain: $x$-values go from 0 to 3 → $ [0, 3] $
#### Range: $y$-values from -3 to 3 → $ [-3, 3] $

✔ Domain: $ [0, 3] $
✔ Range: $ [-3, 3] $

> Not a function (fails vertical line test — e.g., $x=1$ has two $y$-values)

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Problem 7:

Sinusoidal wave-like graph, starts at (-4, 0), peaks at (-2, 2), dips to (0, -1), peaks again at (2, 2), ends at (4, 0)

#### Domain: $x$ from -4 to 4 → $ [-4, 4] $
#### Range: $y$ from -1 to 2 → $ [-1, 2] $

✔ Domain: $ [-4, 4] $
✔ Range: $ [-1, 2] $

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Problem 8:

Curve starting at (-3, -2), increasing, passing through (-1, 0), (1, 2), and going up.

Looks like a parabola opening upward, but only right half.

Starts at $(-3, -2)$, goes up.

#### Domain: $x$ from -3 to ∞? But graph only shows from -3 to 4 → assume closed interval: $ [-3, 4] $
#### Range: $y$ from -2 to 4 → $ [-2, 4] $

But check: lowest $y = -2$, highest $y = 4$?

Wait — at $x = 4$, $y$ appears to be around 4? Yes.

So:

✔ Domain: $ [-3, 4] $
✔ Range: $ [-2, 4] $

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✔ Final Answers Summary:

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#### 1.
- Domain: $ \{-4, -2, 0, 1\} $
- Range: $ \{-4, -1, 1, 3, 4\} $
- Table: see above
- Mapping: see above
- Graph: plot the five points

#### 2.
- Domain: $ \{-3, -1, 0, 2\} $
- Range: $ \{-4, 0, 2, 4, 5\} $
- Table: see above
- Mapping: see above
- Graph: plot the five points

#### 3.
- Domain: $ \{-4, -2, -1, 0, 1, 2\} $
- Range: $ \{-1, 0, 1, 2, 3\} $

#### 4.
- Domain: $ \{-4, -3, -2, -1, 0\} $
- Range: $ \{1, 2, 3\} $

#### 5.
- Domain: $ [-4, 4] $
- Range: $ [-2, 4] $

#### 6.
- Domain: $ [0, 3] $
- Range: $ [-3, 3] $

#### 7.
- Domain: $ [-4, 4] $
- Range: $ [-1, 2] $

#### 8.
- Domain: $ [-3, 4] $
- Range: $ [-2, 4] $

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Let me know if you'd like me to generate a printable version or help with drawing the mappings/graphs!