Domain and Range Mapping Diagrams worksheet for identifying functions.

Let's solve the problem step by step using the definition of a function in the context of mapping diagrams.

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🔷 What is a Function?

A function is a relation where each input (element in the domain) is paired with exactly one output (element in the range).

In other words:
- No element in the domain can map to more than one element in the range.
- But multiple inputs can map to the same output.

This is known as the "vertical line test" for graphs, but in mapping diagrams, we check:
➡️ Does every element on the left (domain) connect to only one element on the right (range)?

If yes, it’s a function.
If any domain element connects to two or more range elements → not a function.

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Now let’s analyze each diagram:

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🔹 Diagram 1:

- Domain: {5, 9, 4}
- Mappings:
- 5 → 8
- 9 → 2
- 4 → 4
- Each input maps to only one output.
✔ Yes, this is a function.

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🔹 Diagram 2:

- Domain: {8, 0, 9}
- Mappings:
- 8 → 0 and 6 (two outputs!)
- 0 → 1 and 4
- 9 → 1
- Both 8 and 0 map to two different values.
✘ No, not a function.

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🔹 Diagram 3:

- Domain: {7, 8, 4, 9}
- Mappings:
- 7 → 5
- 8 → 2
- 4 → 6
- 9 → 9
- All inputs go to one output.
✔ Yes, this is a function.

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🔹 Diagram 4:

- Domain: {4, 3, 1, 6, 2}
- Mappings:
- 4 → 3
- 3 → 8
- 1 → 0
- 6 → 5
- 2 → 6
- Every input has only one arrow.
✔ Yes, this is a function.

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🔹 Diagram 5:

- Domain: {0, 3, 1, 7, 5}
- Mappings:
- 0 → 4 and 2 (two outputs)
- 3 → 0 and 2
- 1 → 6
- 7 → 4
- 5 → 0
- Multiple inputs (0, 3) have more than one output.
✘ No, not a function.

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🔹 Diagram 6:

- Domain: {5, 0, 4}
- Mappings:
- 5 → 8
- 0 → 1
- 4 → 7
- Each input goes to only one output.
✔ Yes, this is a function.

Wait! The answer sheet says "No", but that seems incorrect.

Let’s double-check:
- 5 → 8 ✔
- 0 → 1 ✔
- 4 → 7 ✔

All inputs map to one output — so this should be a function.

But the answer provided is "No" — likely a mistake in the worksheet or misreading.

However, looking at the diagram again:
Is there any ambiguity?

Wait — does 0 map to both 1 and something else? No. Only one arrow from 0 → 1.

So this should be YES.

But the worksheet says "No". Let's recheck carefully.

Ah! Wait — look at the arrow from 4: it goes to 7, but also possibly to another?

Looking at the image:
- 5 → 8
- 0 → 1
- 4 → 7

Only one arrow from each.

So unless there's a typo or misdrawn arrow, this is a function.

But the worksheet says "No" — perhaps an error.

Wait — maybe I'm missing something.

Let me re-express the mappings:

From the diagram:
- 5 → 8
- 0 → 1
- 4 → 7

Each domain element has one arrow → Yes, function.

So the answer key is wrong here.

But since you're asking to solve the problem, we go by logic.

So correct answer for Diagram 6: ✔ Yes

But the given answer is "No" — which is incorrect.

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🔹 Diagram 7:

- Domain: {4, 9, 0}
- Mappings:
- 4 → 7
- 9 → 9
- 0 → 1
- All map to one output.
✔ Yes, function?

Wait — the answer says "No".

But again:
- 4 → 7
- 9 → 9
- 0 → 1

All single outputs → Should be Yes.

But answer says No — contradiction.

Wait — look closely: Is 9 mapped to both 9 and 3?

No — from the diagram:
- 4 → 7
- 9 → 9
- 0 → 1

Only one arrow per input.

So again, this should be a function.

But the answer says "No" — possibly a mistake in the worksheet.

Wait — perhaps the output 9 appears twice? That’s fine — multiple inputs can map to same output.

But no input maps to multiple outputs.

So why "No"?

Wait — maybe 9 in the domain maps to both 9 and 3?

Looking at the diagram:
- From 9: one arrow to 9
- From 4: to 7
- From 0: to 1

Only one arrow from each.

So again, this should be Yes.

But the answer says No — possible error.

Wait — could the arrow from 9 go to both 9 and 3?

Looking at the image carefully:
The diagram shows:
- 4 → 7
- 9 → 9
- 0 → 1

No second arrow from 9.

So still Yes.

But the answer says No — contradiction.

Perhaps the diagram is misread.

Alternatively, maybe the domain includes 9, and 9 maps to 9, but also another arrow?

Wait — in the diagram, the arrow from 9 goes to 9, and also to 3?

Wait — no. It looks like only one arrow from 9.

But in some versions of the worksheet, sometimes arrows are drawn ambiguously.

But based on what's shown: only one arrow per input.

So all three diagrams (6, 7, 8) seem to have issues.

Let’s move to Diagram 8.

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🔹 Diagram 8:

- Domain: {6, 4, 5, 9, 1}
- Mappings:
- 6 → 9
- 4 → 1
- 5 → 7
- 9 → 0
- 1 → 4
- All inputs map to one output.
✔ Yes, function.

But the answer says "No".

Wait — is 4 mapped to both 1 and something else?

No — only one arrow from 4 → 1.

But look: 1 is in the domain, and it maps to 4.

But 4 is also in the domain and maps to 1.

That’s fine — reverse mappings don't matter.

But is any input mapping to two outputs?

No.

So this should be Yes.

But answer says No — again, contradiction.

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🚨 Conclusion: There are errors in the provided answers.

Let’s correctly evaluate each diagram:

| Diagram | Input Mapping | Function? | Reason |
|--------|----------------|-----------|--------|
| 1 | 5→8, 9→2, 4→4 | ✔ Yes | One output per input |
| 2 | 8→0,6; 0→1,4; 9→1 | ✘ No | 8 and 0 map to two outputs |
| 3 | 7→5, 8→2, 4→6, 9→9 | ✔ Yes | One output per input |
| 4 | 4→3, 3→8, 1→0, 6→5, 2→6 | ✔ Yes | One output per input |
| 5 | 0→4,2; 3→0,2; 1→6; 7→4; 5→0 | ✘ No | 0 and 3 map to two outputs |
| 6 | 5→8, 0→1, 4→7 | ✔ Yes | One output per input |
| 7 | 4→7, 9→9, 0→1 | ✔ Yes | One output per input |
| 8 | 6→9, 4→1, 5→7, 9→0, 1→4 | ✔ Yes | One output per input |

But the given answers say:
- 6: No ✘ (should be Yes)
- 7: No ✘ (should be Yes)
- 8: No ✘ (should be Yes)

So three errors in the answer key.

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✔ Final Correct Answers:

1. Yes
2. No
3. Yes
4. Yes
5. No
6. Yes ← (not No)
7. Yes ← (not No)
8. Yes ← (not No)

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🔍 Explanation:

- A function requires that each input (left side) has exactly one output (right side).
- If any input has two or more arrows, it’s not a function.
- In diagrams 6, 7, and 8, each input has only one arrow, so they are functions.

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🛠️ Why the Worksheet Says "No"?

Possibly:
- Misinterpretation of arrows
- Typo in the answer key
- Or a version of the worksheet with different diagrams

But based on the standard interpretation of mapping diagrams, diagrams 6, 7, and 8 are functions.

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✔ Corrected Answer Key:

| Diagram | Function? |
|--------|----------|
| 1 | Yes |
| 2 | No |
| 3 | Yes |
| 4 | Yes |
| 5 | No |
| 6 | Yes |
| 7 | Yes |
| 8 | Yes |

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Let me know if you'd like a visual explanation or clarification on any diagram!