Domain and Range of Graphs worksheet with eight plotted functions on coordinate grids.

Let's solve each of the 8 problems on the worksheet step by step. For each graph, we will determine:

1. Domain: The set of all possible x-values (horizontal extent).
2. Range: The set of all possible y-values (vertical extent).
3. Function?: Whether the graph represents a function (use the Vertical Line Test — if any vertical line intersects the graph more than once, it is not a function).

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1) Graph 1

- Graph Description: A single curve from (0, 4) to (2, 0), decreasing, with a solid dot at (0, 4) and an open circle at (2, 0).
- Domain: x-values from 0 to 2 → [0, 2)
- Range: y-values from 0 to 4 → [0, 4]
- Function?: Yes – passes vertical line test.
- ✔ Answer:
- Domain: [0, 2)
- Range: [0, 4]
- Function: Yes

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2) Graph 2

- Graph Description: A downward-opening parabola arc from (-3, 0) to (3, 0), peaking at (0, 3). Solid dots at endpoints.
- Domain: x-values from -3 to 3 → [-3, 3]
- Range: y-values from 0 to 3 → [0, 3]
- Function?: Yes – every x has only one y.
- ✔ Answer:
- Domain: [-3, 3]
- Range: [0, 3]
- Function: Yes

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3) Graph 3

- Graph Description: A wavy curve starting at (-3, -2), going up to (0, 2), down to (2, -2), then up again. Open circle at (-3, -2), closed at (2, -2), and a closed point at (0, 2).
- Domain: x-values from -3 to 2 → [-3, 2]
- Range: y-values from -2 to 2 → [-2, 2]
- Function?: Yes – no vertical line crosses twice.
- ✔ Answer:
- Domain: [-3, 2]
- Range: [-2, 2]
- Function: Yes

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4) Graph 4

- Graph Description: A U-shaped curve opening to the right, from (-2, -2) to (2, 2), but not continuous. It looks like two parts: left side from (-2, -2) to (0, 0), and right side from (0, 0) to (2, 2), but only one part is drawn: from (-2, -2) to (0, 0) with a solid dot at (-2, -2), open at (0, 0); and another segment from (0, 0) to (2, 2), open at (0, 0), solid at (2, 2). So it’s a V-shape opening to the right, but only the left half is shown?
Wait — actually, looking closely: it seems like a parabola opening to the right, with points from (-2, -2) to (2, 2), but only the left half is drawn? No — let's recheck.

Actually, this graph shows two separate curves:
- One from (-2, -2) to (0, 0): solid at (-2, -2), open at (0, 0)
- One from (0, 0) to (2, 2): open at (0, 0), solid at (2, 2)

So it’s like a "V" shape pointing to the right, but split at origin.

But wait — each x-value has two y-values? Let's check:

For example, x = 0 appears in both segments, but only at (0,0), which is open in both — so no point at x=0.

But for x = 1, there are two points? Wait, no — the graph shows only one curve: a semicircle or curved path from (-2,-2) to (2,2) passing through (0,0), but drawn as two parts?

Actually, upon closer inspection, it looks like a parabola-like shape symmetric about the origin, but only one branch is drawn: from (-2, -2) to (2, 2) through (0,0), but with open circles at (0,0) on both ends? No — it's a single continuous curve from (-2, -2) to (2, 2), passing through (0,0), with closed dot at (-2,-2), open dot at (2,2), and open dot at (0,0)?

Wait — no. Looking carefully:

It's a curve that starts at (-2, -2) with a solid dot, goes upward to (0, 0) with an open dot, then continues to (2, 2) with a solid dot? But that would be two segments.

But the x-values from -2 to 2 are covered, and for each x, there's only one y — so it could be a function.

But wait — if it's a curve from (-2,-2) to (2,2) passing through (0,0), and it's smooth, then it might be a function.

But here’s the key: at x = 0, there is an open circle, meaning the point (0,0) is not included.

But the curve goes from (-2,-2) to (0,0) (open), then from (0,0) (open) to (2,2) (solid). So it's a continuous curve with a gap at (0,0)? That doesn't make sense.

Wait — actually, the graph is likely a function defined piecewise.

Looking at standard interpretations: This graph appears to be a V-shape rotated 45 degrees, but more likely, it's a parabola opening to the right, but only the left half is drawn? No.

Actually, after careful analysis, it looks like a function where:
- From x = -2 to x = 0, y increases from -2 to 0
- From x = 0 to x = 2, y increases from 0 to 2
- But at x = 0, there is an open circle, meaning (0,0) is not included
- However, the curve approaches (0,0) from both sides

But since (0,0) is not included, and the curve is smooth, it's still a function as long as each x has one y.

But wait — if the curve is continuous and increasing from (-2,-2) to (2,2), and (0,0) is not included, then x=0 has no value, so it's not defined at x=0.

But the graph shows a break at x=0.

So:
- Domain: [-2, 0) ∪ (0, 2] → because x=0 is excluded
- Range: [-2, 0) ∪ (0, 2] → same
- But is it a function? Yes — for each x in domain, only one y

✔ Answer:
- Domain: [-2, 0) ∪ (0, 2]
- Range: [-2, 0) ∪ (0, 2]
- Function: Yes

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5) Graph 5

- Graph Description: A line segment from (-4, 2) to (-1, -1), then a horizontal line from (-1, -1) to (0, -1), then a line from (0, -1) to (1, 0). Open circle at (-4,2), closed at (1,0), and closed at (-1,-1), open at (0,-1)? Wait — no.

Actually:
- Starts at (-4, 2) with open circle → not included
- Goes down to (-1, -1) with closed circle → included
- Then horizontal to (0, -1) with open circle → not included
- Then up to (1, 0) with closed circle → included

So:
- Domain: x from -4 to 1, but excluding -4 and 0? Wait — from -4 to -1 (inclusive), then from -1 to 0 (but open at 0), then from 0 to 1 (but 0 is open), so x=0 is missing

Wait — no: the horizontal segment is from (-1, -1) to (0, -1), with open circle at (0,-1) → so x=0 is not included in that segment.

Then the next segment starts at (0, -1) — but that point is open, so no point at (0,-1).

So the graph has:
- From x = -4 to x = -1: line from (-4,2) to (-1,-1), open at (-4), closed at (-1)
- From x = -1 to x = 0: horizontal line from (-1,-1) to (0,-1), closed at (-1), open at (0)
- From x = 0 to x = 1: line from (0,-1) to (1,0), open at (0), closed at (1)

So x=0 is not included in any segment.

Thus:
- Domain: [-4, -1] ∪ [-1, 0) ∪ (0, 1] = [-4, 0) ∪ (0, 1]
- Range: y-values from -1 to 2 → [-1, 2]
- Is it a function? Yes — each x has only one y

✔ Answer:
- Domain: [-4, 0) ∪ (0, 1]
- Range: [-1, 2]
- Function: Yes

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6) Graph 6

- Graph Description: A straight line from (-3, -2) to (3, 4), with solid dots at both ends.
- Domain: x from -3 to 3 → [-3, 3]
- Range: y from -2 to 4 → [-2, 4]
- Function?: Yes — straight line, passes vertical line test
- ✔ Answer:
- Domain: [-3, 3]
- Range: [-2, 4]
- Function: Yes

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7) Graph 7

- Graph Description: A diagonal line from (-4, 2) to (0, -2), with open circle at (-4,2), closed at (0,-2)
- Domain: x from -4 to 0 → (-4, 0]
- Range: y from -2 to 2 → [-2, 2)
- Function?: Yes — straight line, one y per x
- ✔ Answer:
- Domain: (-4, 0]
- Range: [-2, 2)
- Function: Yes

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8) Graph 8

- Graph Description: A curve that starts at (-3, 0) with open circle, goes up to (0, 3), then down to (3, -3) with open circle. It's a U-shaped curve but inverted — like a parabola opening downward, but only the right half? No — it's not a function.

Wait — look at x = 0: it has two y-values — one at (0,3) and possibly others?

No — the curve goes from (-3,0) to (0,3), then from (0,3) to (3,-3). But at x=0, there is only one point — (0,3) — so maybe it's a function?

Wait — no: the graph shows a loop or non-function.

Actually, it's a parabola opening to the right, but with a double curve — like a sideways parabola.

But more precisely: it looks like a function — from x=-3 to x=3, but at x=0, only one point.

Wait — no: the curve goes from (-3,0) to (0,3), then from (0,3) to (3,-3). So it's a V-shape pointing downward? Or a zigzag?

But at x=1, there is only one y-value? Let's see.

Actually, the graph shows a single continuous curve from (-3,0) to (3,-3), passing through (0,3). But does it go back?

Wait — the curve from (-3,0) to (0,3) is increasing, then from (0,3) to (3,-3) is decreasing — but that’s fine.

But is it a function? Yes — for each x, only one y.

But wait — the shape suggests it may be a function.

However, upon closer inspection, it looks like a sideways parabola — but no, it’s a piecewise linear curve.

But the issue is: does it pass the vertical line test?

At x=1, there is only one point — yes.

But the curve goes from (-3,0) to (0,3), then from (0,3) to (3,-3). So it’s a single path — a broken line.

But at x=0, it’s (0,3) — only one point.

So it is a function.

But wait — the graph shows a loop? No — it's a smooth curve from left to right.

Wait — actually, looking at the graph, it might be a parabola opening to the right, but with two branches — but in this case, it’s not.

Alternatively, it might be a function with a cusp or turn.

But the key is: does any vertical line cross it more than once?

From the description: it goes from (-3,0) to (0,3), then down to (3,-3). So it's a single path — a V-shape turned.

But it's not a function because at some x-values, there are two y-values?

Wait — no — it’s a continuous curve from left to right — so it’s a function.

But let’s think: if it goes from (-3,0) to (0,3), then from (0,3) to (3,-3), that’s fine — it’s a function.

But the problem is: the graph shows a loop — no, it doesn’t.

Wait — actually, upon close inspection, it might be a sideways parabola — but the way it's drawn, it’s not.

Alternatively, it could be a function — but let's check the vertical line test.

If I draw a vertical line at x=1, it hits only one point — yes.

So it is a function.

But wait — the graph has a sharp turn at (0,3), but that’s okay.

So:
- Domain: x from -3 to 3 → [-3, 3]
- Range: y from -3 to 3 → [-3, 3]
- Function: Yes

Wait — but the endpoints:
- At (-3,0): open circle → not included
- At (3,-3): open circle → not included
- So domain: (-3, 3)
- Range: (-3, 3) — but y reaches 3 at (0,3), which is closed → so y=3 is included

So:
- Domain: (-3, 3)
- Range: [-3, 3]
- Function: Yes

✔ Answer:
- Domain: (-3, 3)
- Range: [-3, 3]
- Function: Yes

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Final Answers:

| # | Domain | Range | Function? |
|---|--------|-------|----------|
| 1 | [0, 2) | [0, 4] | Yes |
| 2 | [-3, 3] | [0, 3] | Yes |
| 3 | [-3, 2] | [-2, 2] | Yes |
| 4 | [-2, 0) ∪ (0, 2] | [-2, 0) ∪ (0, 2] | Yes |
| 5 | [-4, 0) ∪ (0, 1] | [-1, 2] | Yes |
| 6 | [-3, 3] | [-2, 4] | Yes |
| 7 | (-4, 0] | [-2, 2) | Yes |
| 8 | (-3, 3) | [-3, 3] | Yes |

> Note: All graphs appear to be functions based on the vertical line test.

Let me know if you'd like a visual explanation or correction for any specific graph!