Math worksheet titled 'Relations & Functions' asking students to determine if various representations are functions and to list their domain and range.

Let's go through each of the given problems and determine whether each relation is a function, and if so, identify its domain and range.

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1. Table:

```
x | y
-6 | 3
-5 | 2
-4 | 3
-3 | 7
-2 | 8
```

- Is it a function?
Yes. Each input (x-value) maps to exactly one output (y-value). Even though y = 3 appears twice (for x = -6 and x = -4), that’s allowed in functions. The key is that no x-value repeats with different y-values. Here, all x-values are unique.

- Domain: All x-values → {-6, -5, -4, -3, -2}
- Range: All y-values → {3, 2, 7, 8}

✔ Function: Yes
Domain: {-6, -5, -4, -3, -2}
Range: {2, 3, 7, 8}

---

2. Graph (Scatter Plot):

This graph shows points:
(-2, 0), (-1, 1), (0, 2), (1, 3), (2, 4)

- Vertical Line Test: No vertical line passes through more than one point. So, it's a function.

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

✔ Function: Yes
Domain: {-2, -1, 0, 1, 2}
Range: {0, 1, 2, 3, 4}

---

3. Mapping Diagram:

Left side (input): -1, 0, 1, 2
Right side (output): -2, -1, 0, 1

Arrows:
- -1 → -2
- 0 → -1
- 1 → 0
- 2 → 1

Each input has exactly one output, and no input maps to multiple outputs.

✔ Function: Yes
Domain: {-1, 0, 1, 2}
Range: {-2, -1, 0, 1}

---

4. Graph (Line on coordinate plane):

This is a straight line passing through the origin with a positive slope.

- Vertical Line Test: Any vertical line intersects the graph at most once → Yes, it's a function.

- Domain: All real numbers (the line extends infinitely left and right) → $(-\infty, \infty)$
- Range: All real numbers (line goes up and down infinitely) → $(-\infty, \infty)$

✔ Function: Yes
Domain: $(-\infty, \infty)$
Range: $(-\infty, \infty)$

---

5. Table:

```
x | y
-3 | 4
-2 | 4
-1 | 4
0 | 4
1 | 4
```

All x-values map to y = 4. Each input has one output. Even though all outputs are the same, this is still a function.

✔ Function: Yes
Domain: {-3, -2, -1, 0, 1}
Range: {4}

---

6. Mapping Diagram:

Input: -3, -2, -1, 0, 1
Output: 1, 2, 3, 4, 5

But look closely:
- -3 → 1 and 2 → ✘ Not a function!
- -2 → 3 and 4 → ✘
- -1 → 5 → okay
- 0 → 1 → okay
- 1 → 2 → okay

Wait — some inputs have two outputs. For example, -3 maps to both 1 and 2.

✘ Function: No
(Not a function because one input maps to multiple outputs.)

- Domain: {-3, -2, -1, 0, 1}
- Range: {1, 2, 3, 4, 5}
But since it's not a function, we don't define domain/range as per function rules.

So:

✘ Function: No
Domain: {-3, -2, -1, 0, 1}
Range: {1, 2, 3, 4, 5}

---

7. Set of Ordered Pairs:

{(−4, −2), (−3, 0), (0, 1), (3, 4)}

- Check for repeated x-values: All x-values are unique: -4, -3, 0, 3 → no duplicates.

✔ Function: Yes
Domain: {-4, -3, 0, 3}
Range: {-2, 0, 1, 4}

---

8. Set of Ordered Pairs:

{(−6, 4), (−4, −2), (2, −1), (−4, 8), (0, 0)}

- Look at x = -4: It appears twice: once with y = -2, once with y = 8 → same input, different outputs → not a function

✘ Function: No
Domain: {-6, -4, 2, 0}
Range: {4, -2, -1, 8, 0} → but again, since not a function, we just list values.

---

✔ Final Answers:

| Problem | Function? | Domain | Range |
|--------|-----------|--------|-------|
| 1 | Yes | {-6, -5, -4, -3, -2} | {2, 3, 7, 8} |
| 2 | Yes | {-2, -1, 0, 1, 2} | {0, 1, 2, 3, 4} |
| 3 | Yes | {-1, 0, 1, 2} | {-2, -1, 0, 1} |
| 4 | Yes | $(-\infty, \infty)$ | $(-\infty, \infty)$ |
| 5 | Yes | {-3, -2, -1, 0, 1} | {4} |
| 6 | No | {-3, -2, -1, 0, 1} | {1, 2, 3, 4, 5} |
| 7 | Yes | {-4, -3, 0, 3} | {-2, 0, 1, 4} |
| 8 | No | {-6, -4, 2, 0} | {4, -2, -1, 8, 0} |

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🔍 Key Concepts:

- A function must assign each input to exactly one output.
- Use the vertical line test for graphs.
- In tables or sets of ordered pairs, no x-value should repeat with different y-values.
- In mapping diagrams, each input arrow must go to only one output.

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