Relations and Functions Notes and Worksheets - Lindsay Bowden - Free Printable

Let's go through each of the 8 problems and determine whether each relation is a function.

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🔹 Definition: What is a Function?

A function is a relation where each input (x-value) has exactly one output (y-value).
In other words:
- No x-value can be paired with more than one y-value.
- If any input appears more than once with different outputs, it’s not a function.

We'll check each problem accordingly.

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✔ Problem 1:

| x | y |
|---|---|
| 5 | 1 |
| 5 | 2 |
| 5 | 3 |
| 5 | 4 |

- The input 5 appears four times, each time with a different output.
- So, one input maps to multiple outputs → ✘ Not a function

> Answer: Not a function

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✔ Problem 2:

Input → Output:
- 6 → 9
- 8 → 11
- 11 → 15
- 14 → 20

Each input goes to only one output, and no input repeats.

> Answer: ✔️ Function

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✔ Problem 3:

Input → Output:
- -5 → 2
- -2 → 8
- 1 → 7
- 0 → 6

Wait — look closely:
- Input 1 connects to both 7 and 6?
- Actually, from the diagram:
- 1 → 7
- 1 → 6? Wait — does it?

Looking at the arrows:
- -5 → 2
- -2 → 8
- 1 → 7
- 0 → 6

But also:
- -5 → 6? (No, arrow from -5 to 2 only)
Wait — actually, let's trace:
- -5 → 2
- -2 → 8
- 1 → 7
- 0 → 6

And:
- -5 → 6? No, that’s not shown.
Wait — there are two lines from 1: one to 7, one to 6?

Yes! Look carefully:
- From 1, two arrows go out: one to 7, one to 6 → That means input 1 has two outputs → ✘

> Answer: ✘ Not a function

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✔ Problem 4:

List of ordered pairs:
- (-9, 4)
- (2, 8)
- (0, 1)
- (6, 5)
- (2, 3)

Now check for repeated x-values:
- x = 2 appears twice: (2,8) and (2,3) → same input, different outputs → ✘

> Answer: ✘ Not a function

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✔ Problem 5:

| x | 6 | 8 | 10 | 12 | 14 |
|---|---|---|----|----|----|
| y |-4 | 2 | 6 |-4 | 11 |

Check if any x-value repeats:
- All x-values are unique: 6, 8, 10, 12, 14 → all different

Even though y = -4 appears twice (for x=6 and x=12), that’s allowed in functions — outputs can repeat, but inputs must be unique.

> Answer: ✔️ Function

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✔ Problem 6:

Input → Output:
- 3 → 2
- 6 → 4
- 9 → 6
- 12 → 4

Now check:
- Each input has only one output.
- Input 6 → 4
- Input 12 → 4 → okay, same output is fine.
- No input repeats.

> Answer: ✔️ Function

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✔ Problem 7:

Ordered pairs:
- (-11, 14)
- (-13, 14)
- (3, 12)
- (1, 14)
- (8, 11)

All x-values are different: -11, -13, 3, 1, 8 → all unique.

Even though y = 14 appears multiple times, that's acceptable.

> Answer: ✔️ Function

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✔ Problem 8:

| x | y |
|---|---|
| -9 | 4 |
| -8 | 3 |
| -8 | 1 |
| -6 | 0 |
| -4 | -3 |

Look at x = -8 → appears twice:
- (-8, 3)
- (-8, 1)

Same input, different outputs → ✘

> Answer: ✘ Not a function

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✔ Final Answers Summary:

| Problem | Is it a Function? |
|--------|-------------------|
| 1 | ✘ No |
| 2 | ✔️ Yes |
| 3 | ✘ No |
| 4 | ✘ No |
| 5 | ✔️ Yes |
| 6 | ✔️ Yes |
| 7 | ✔️ Yes |
| 8 | ✘ No |

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📝 Explanation Recap:

- A relation is a function only if each input (x) has exactly one output (y).
- Repeated inputs with different outputs → not a function.
- Repeated outputs are fine — that’s allowed.

Let me know if you'd like this formatted as a worksheet answer key!