Math worksheet with graphing problems and solutions.

Let’s go through each problem step by step.

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Top Left: Mapping Diagram (Two Ovals with Arrows)
Left oval: {1, 2, 3}
Right oval: {4, 5, 6}
Arrows:
- 1 → 4
- 2 → 5
- 3 → 6

Each input has exactly one output → Function
No two inputs map to same output → One-to-One
All outputs are used → Onto
So it’s a bijection, but since the question likely asks for function type:
✔ Function, One-to-One, Onto

But looking at label under diagram: “N/P” — probably means “Not a Function / Not a Relation”? Wait — no, this *is* a function. Maybe N/P is mislabeled? Let’s check others.

Actually, looking again — maybe “N/P” stands for “Not a Function / Partial”? But in standard terms, this is a function.

Wait — perhaps the labels under diagrams are answers already given? The user might be asking to verify or solve based on those.

Looking at top right:

Top Right: Graph with points and arrows between ovals
Left oval: {-2, -1, 0, 1, 2}
Right oval: {0, 1, 4}
Arrows:
- -2 → 4
- -1 → 1
- 0 → 0
- 1 → 1
- 2 → 4

Each input maps to one output → Function
But multiple inputs map to same output (e.g., -2 and 2 both → 4) → Not One-to-One
All outputs used? 0,1,4 — yes → Onto if codomain is {0,1,4}
Label says “N/S” — maybe “Not Surjective”? But it is surjective onto its range.

Hmm — perhaps we need to interpret what the student is being asked.

Looking at bottom row:

Bottom Left: Line graph with point (-2, -2), (-1, -1), etc.
It’s a straight line through origin with slope 1 → y = x
Domain shown: from x=-2 to x=2? But arrow goes beyond → infinite domain?
Point labeled: (-2, -2), (-1, -1), (0,0), (1,1), (2,2) — all on y=x
This is a function, one-to-one, onto if domain and codomain are all reals.
Label: “Yes” — probably meaning “Is it a function?” → Yes

Bottom Middle: Circle centered at origin, radius 1
Equation: x² + y² = 1
Fails vertical line test → Not a function
Label: “No” — correct

Bottom Right: Sine-like curve from (-π/2, -1) to (π/2, 1)
Looks like y = sin(x) restricted to [-π/2, π/2]
Passes vertical line test → Function
Also passes horizontal line test in this interval → One-to-One
Range is [-1,1], which matches codomain if defined that way → Onto
Label: “Yes” — probably “Is it a function?” → Yes

Now back to top left mapping:
Inputs: 1,2,3 → Outputs: 4,5,6 — bijective → definitely a function → should be “Yes” but labeled “N/P” — maybe typo?

Top middle: coordinate plane with scattered points:
Points: (1,2), (2,4), (3,6), (4,8) — looks like y=2x
But also point (2,3)? Wait — let me see:
In top middle grid:
At x=1, y=2
x=2, y=4 and also y=3? Oh! There’s a point at (2,3) AND (2,4) — so same x, different y → NOT a function
Label: “N/S” — maybe “Not a Function / Something”

Top right mapping: as above — is a function, not one-to-one, but onto if codomain is {0,1,4} — label “N/S” — confusing.

Perhaps the task is to determine whether each relation is a function, and if so, whether it’s one-to-one and/or onto.

Let me reorganize:

We have 6 diagrams:

1. Top-left mapping: 1→4, 2→5, 3→6 → Function, Injective, Surjective → Bijection
2. Top-middle graph: Points include (2,3) and (2,4) → Not a function
3. Top-right mapping: -2→4, -1→1, 0→0, 1→1, 2→4 → Function, Not injective (since -2 and 2 both map to 4), Surjective if codomain is {0,1,4}
4. Bottom-left line: y=x → Function, Injective, Surjective (if domain=codomain=R)
5. Bottom-middle circle: x²+y²=1 → Not a function
6. Bottom-right sine curve: y=sin(x) on [-π/2, π/2] → Function, Injective, Surjective onto [-1,1]

The labels under them seem to be partial answers or hints:

- Top-left: “N/P” — maybe “Not a Problem”? Unlikely. Perhaps “Non-Partial”? Doesn’t fit.
- Top-middle: “N/S” — “Not a Function / Surjective”? But it’s not a function.
- Top-right: “N/S” — same issue.
- Bottom-left: “Yes” — is it a function? Yes.
- Bottom-middle: “No” — is it a function? No.
- Bottom-right: “Yes” — is it a function? Yes.

Ah! Probably the main question for each is: “Is this relation a function?”

Then secondary questions might be about one-to-one or onto, but the primary label seems to be answering “function or not”.

So:

1. Top-left mapping: Is it a function? → YES (each input has one output) → but labeled “N/P” — contradiction? Unless “N/P” means something else.

Wait — look at the very top: there’s a small diagram with C₁(3) and C₂(5) — maybe context?

Another idea: perhaps “N/P” means “Not a Function because Partial” — but in discrete math, a partial function allows some inputs unmapped, but here all are mapped.

I think there might be a mislabeling in the worksheet, or I’m misinterpreting.

Let’s focus on clear cases:

- Any graph or mapping where an input has more than one output → NOT a function.
- Otherwise → IS a function.

Apply that:

Diagram 1 (top-left mapping): Each input (1,2,3) has exactly one output → FUNCTION → Answer: YES

Diagram 2 (top-middle graph): Point at x=2 has two y-values (y=3 and y=4) → NOT A FUNCTION → Answer: NO

Diagram 3 (top-right mapping): Each input (-2,-1,0,1,2) has exactly one output → FUNCTION → Answer: YES

Diagram 4 (bottom-left line): Straight line, passes vertical line test → FUNCTION → Answer: YES

Diagram 5 (bottom-middle circle): Fails vertical line test (e.g., x=0 has y=1 and y=-1) → NOT A FUNCTION → Answer: NO

Diagram 6 (bottom-right sine curve): Only one y per x in the drawn portion → FUNCTION → Answer: YES

Now compare to labels provided in image:

- Top-left: labeled “N/P” — should be YES → discrepancy
- Top-middle: labeled “N/S” — should be NO → matches if “N” means No
- Top-right: labeled “N/S” — should be YES → discrepancy
- Bottom-left: “Yes” → matches
- Bottom-middle: “No” → matches
- Bottom-right: “Yes” → matches

So only top-left and top-right are mismatched with my analysis.

Re-examining top-left: Inputs 1,2,3; outputs 4,5,6 — all good. Why “N/P”?

Unless... is the codomain larger? But in mapping diagrams, if outputs are only those shown, it’s fine.

Perhaps “N/P” means “Not a Proper Function” — but that’s not standard.

Another thought: in some contexts, if the mapping is not total (some inputs missing), but here all inputs are covered.

I think the safest approach is to go by definition:

A relation is a function if every element in the domain maps to exactly one element in the codomain.

For mappings: check if any input has zero or multiple arrows out.

For graphs: check vertical line test.

So final determination:

1. Top-left mapping: FUNCTION → YES
2. Top-middle graph: NOT FUNCTION → NO (because x=2 has two y's)
3. Top-right mapping: FUNCTION → YES
4. Bottom-left line: FUNCTION → YES
5. Bottom-middle circle: NOT FUNCTION → NO
6. Bottom-right curve: FUNCTION → YES

But since the image has labels, and the user might want verification, I’ll assume the task is to say for each whether it’s a function, and the “Final Answer” should list them.

Perhaps the “N/P”, “N/S” are distractors or errors.

Given that, I’ll provide the correct classification.

To match format, since ExplainAnswer is false, I won’t explain, but I did reason internally.

Final Answer should be the answers for each diagram.

How to present? Probably in order.

Since the image has 6 diagrams, and likely expects 6 answers.

From left to right, top to bottom:

Row 1:
- Diagram 1: Function? Yes
- Diagram 2: Function? No
- Diagram 3: Function? Yes

Row 2:
- Diagram 4: Function? Yes
- Diagram 5: Function? No
- Diagram 6: Function? Yes

So sequence: Yes, No, Yes, Yes, No, Yes

But let’s confirm diagram 2: in the grid, at x=2, there are two points: one at (2,3) and one at (2,4)? Looking closely at the image description — yes, in top-middle, there is a point at (2,3) and another at (2,4), so definitely not a function.

Diagram 3: all inputs have single output — function.

So I think that’s solid.

Final Answer:
Yes, No, Yes, Yes, No, Yes