Worksheet for determining whether given relations are functions.

Let's solve this step-by-step. The task is to determine whether each of the 20 given relations is a function or not.

🔍 Definition: What is a Function?

A relation is a function if and only if each input (x-value) corresponds to exactly one output (y-value).

This means:
- No x-value can be paired with more than one y-value.
- We can use the Vertical Line Test for graphs: if any vertical line intersects the graph more than once, it’s not a function.

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We'll go through each item from 1 to 20:

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✔ 1. Table:

```
x | y
5 | 3
-1| 2
0 | 1
5 | 7
```
- x = 5 appears twice with different y-values (3 and 7) → Not a function

👉 NO

---

✔ 2. Set of ordered pairs:

{(2,4), (4,6), (6,8), (8,10)}
- All x-values are unique → Yes, each x has one y

👉 YES

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✔ 3. Graph (parabola opening right):

- This is a sideways parabola (like x = y²).
- Vertical line test fails: some vertical lines cross it twice.

👉 NO

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✔ 4. Mapping diagram (circle to circle):

- Left side: {1, 2, 3} → Right side: {1}
- Each element on the left maps to one element on the right.
- Even though multiple inputs map to same output, that’s allowed in functions.

👉 YES

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✔ 5. Graph (zigzag line):

- It looks like a piecewise linear graph.
- Passes vertical line test: no vertical line hits more than one point.

👉 YES

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✔ 6. Arrow diagram:

- Input: {6, 5, 2, 1}, Output: {4, -1}
- Check: 6→4, 5→-1, 2→4, 1→-1
- Each input maps to exactly one output → valid

👉 YES

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✔ 7. Graph (sideways parabola):

- Opens to the right → same as #3.
- Fails vertical line test → multiple y-values for same x.

👉 NO

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✔ 8. Scatter plot on grid:

- Points: (-2,0), (-1,1), (0,2), (1,1), (2,0)
- All x-values are unique → no repeated x → each x has one y

👉 YES

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✔ 9. Table:

```
x | y
-3| 2
0 | 2
1 | 2
2 | 2
```
- All x-values are unique → each x maps to one y

👉 YES

---

✔ 10. Set: {(−2, −1), (3, 7), (−2, 4)}

- x = -2 appears twice with different y-values: -1 and 4 → conflict

👉 NO

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✔ 11. Set: {(-1,3), (0,-4), (1,-7)}

- All x-values unique → each x has one y

👉 YES

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✔ 12. Graph (wavy horizontal line):

- Looks like a sine wave but horizontal? Wait — actually, it’s a horizontal wave (oscillating up/down).
- But if it’s a standard oscillating curve like y = sin(x), it passes vertical line test.

Wait — looking closely: it's a wavy line going left to right, so likely y = f(x).

Even though it goes up and down, each x has one y → passes vertical line test.

👉 YES

---

✔ 13. Mapping diagram:

- Left set: {1, 2, 3}, Right: {5, 6, 7}
- Arrows: 1→5, 2→6, 3→7 → each input maps to one output

👉 YES

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✔ 14. Graph (sinusoidal wave):

- Like a sine wave: smooth oscillation.
- Every x has one y → passes vertical line test.

👉 YES

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✔ 15. Graph (straight diagonal line):

- Clearly a straight line, increasing.
- Passes vertical line test.

👉 YES

---

✔ 16. Table:

```
x | y
2 | 4
4 | 7
2 | 8
1 | 12
```
- x = 2 appears twice: y = 4 and y = 8 → same input, two outputs

👉 NO

---

✔ 17. Scatter plot (dots):

- Points: (1,1), (2,2), (3,3), (4,4), (5,5)? Wait — let's see:
- Actually, points seem to be at (1,1), (2,2), (3,3), (4,4), (5,5), etc.
- All x-values are distinct → one y per x

👉 YES

---

✔ 18. Set: {(2,4), (6,6), (12,9)}

- All x-values are unique → one output per input

👉 YES

---

✔ 19. Mapping diagram (two circles):

- Left: {1, 2, 3}, Right: {4, 5, 6}
- Arrows: 1→4, 2→5, 3→6 → all inputs map to one output

👉 YES

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✔ 20. Graph (cross-shaped dots):

- Dots form a cross: centered at origin.
- Look at x = 0: multiple y-values (e.g., (0, -3), (0, -2), (0, -1), (0, 0), (0, 1), (0, 2), (0, 3))
- So x = 0 maps to many y-values → fails vertical line test

👉 NO

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✔ Final Answers:

| # | Answer |
|---|--------|
| 1 | NO |
| 2 | YES |
| 3 | NO |
| 4 | YES |
| 5 | YES |
| 6 | YES |
| 7 | NO |
| 8 | YES |
| 9 | YES |
|10 | NO |
|11 | YES |
|12 | YES |
|13 | YES |
|14 | YES |
|15 | YES |
|16 | NO |
|17 | YES |
|18 | YES |
|19 | YES |
|20 | NO |

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✔ Summary of YES/NO:

- YES: 2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19
- NO: 1, 3, 7, 10, 16, 20

> ✔ Total YES: 14
> ✘ Total NO: 6

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