Algebra 1 Worksheets | Domain and Range Worksheets - Free Printable

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 circle at (2, -2). Ends at (3, 2) with open circle.
- Domain: x-values from -3 to 3 → (-3, 3]
- Range: y-values from -2 to 2 → [-2, 2]
- Function?: Yes – no vertical line crosses more than once.
- ✔ Answer:
- Domain: (-3, 3]
- Range: [-2, 2]
- Function: Yes

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4) Graph 4

- Graph Description: A curved path from (-2, 0) to (0, 2), then back down to (-1, 0), forming a "U" shape but not symmetric. Closed dot at (-2, 0), open circle at (-1, 0), closed dot at (0, 2). Also a point at (-1, -2)? Wait — actually, it looks like two separate parts: from (-2, 0) to (0, 2) and from (-1, -2) to (-1, 0)?
- Actually, looking closely: It’s a piecewise curve:
- From (-2, 0) to (0, 2) — increasing
- Then from (-1, -2) to (-1, 0) — vertical segment?
- But wait — there’s a vertical line at x = -1 from y = -2 to y = 0.
- That means multiple y-values for x = -1 → fails vertical line test.

Wait! Let's re-analyze:

- At x = -1, there are two points: (-1, -2) and (-1, 0)? No — the graph shows a vertical segment from (-1, -2) to (-1, 0) — that’s a vertical line, so multiple outputs for same input.

So:
- Function?: ✘ No — fails vertical line test at x = -1.

But let's confirm the graph:
- Left part: from (-2, 0) to (0, 2), increasing
- Right part: from (-1, -2) to (-1, 0), vertical line segment
- So at x = -1, y goes from -2 to 0 → multiple y-values → not a function.

Also, the domain includes x = -1 twice.

✔ Answer:
- Domain: [-2, 0] ∪ {-1} → but since it's continuous, better to write as: [-2, 0] (but the vertical line is at x = -1, which is included in this interval)

Wait — actually, the graph seems to have:
- A curve from (-2, 0) to (0, 2)
- And a vertical line from (-1, -2) to (-1, 0)

But x = -1 is already in [-2, 0], so this is a vertical segment at x = -1, meaning not a function.

So:
- Domain: x-values: from -2 to 0 (the horizontal span) → [-2, 0]
- Range: y-values: from -2 to 2 → [-2, 2]
- Function?: ✘ No — because of vertical segment at x = -1

✔ Answer:
- Domain: [-2, 0]
- Range: [-2, 2]
- Function: No

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5) Graph 5

- Graph Description: Two lines forming a "V" shape upside down. Starts at (-4, 3) with open circle, goes down to (-1, 0), then up to (0, -1) with open circle.
- Points:
- (-4, 3): open circle
- (-1, 0): closed
- (0, -1): open circle
- So:
- Left segment: from (-4, 3) to (-1, 0), open at left, closed at right
- Right segment: from (-1, 0) to (0, -1), closed at left, open at right
- Domain: x-values from -4 to 0 → (-4, 0)
- Range: y-values from -1 to 3 → [-1, 3)
- Max y = 3 (not reached), min y = -1 (not reached) → so range is (-1, 3)? Wait:
- At (-1, 0): y = 0
- At (-4, 3): y = 3, but open → not included
- At (0, -1): y = -1, open → not included
- So lowest y is approaching -1, highest approaching 3
- But does it reach values between?

Actually, the graph goes from y=3 (open) down to y=0, then down to y=-1 (open). So:
- Maximum y: approaches 3 → y < 3
- Minimum y: approaches -1 → y > -1
- But y=0 is achieved at (-1, 0)

So range: (-1, 3)

But is that correct? Let's see:
- From (-4, 3) to (-1, 0): y decreases from 3 to 0 → so y ∈ (0, 3)
- From (-1, 0) to (0, -1): y decreases from 0 to -1 → y ∈ (-1, 0)
- Combined: y ∈ (-1, 3), excluding endpoints

Yes.

- Domain: x from -4 to 0, open at both ends → (-4, 0)
- Function?: Yes — each x has only one y
- ✔ Answer:
- Domain: (-4, 0)
- Range: (-1, 3)
- Function: Yes

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6) Graph 6

- Graph Description: A straight line from (-3, -3) to (3, 3), passing through origin. Both endpoints are solid dots.
- Domain: x from -3 to 3 → [-3, 3]
- Range: y from -3 to 3 → [-3, 3]
- Function?: Yes — straight line, passes vertical line test
- ✔ Answer:
- Domain: [-3, 3]
- Range: [-3, 3]
- Function: Yes

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7) Graph 7

- Graph Description: A line segment from (-4, 3) to (0, -1). Open circle at (-4, 3), closed circle at (0, -1).
- Domain: x from -4 to 0 → (-4, 0]
- Range: y from -1 to 3 → [-1, 3)
- Because y starts at 3 (open), ends at -1 (closed)
- Function?: Yes — straight line, one y per x
- ✔ Answer:
- Domain: (-4, 0]
- Range: [-1, 3)
- Function: Yes

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8) Graph 8

- Graph Description: A curve from (-3, -3) to (2, 3), but it's not a function. It has a loop or U-shape that turns back.
- Specifically: starts at (-3, -3), goes up to (0, 3), then down to (2, -3), but wait — at x=0, it reaches y=3, then goes down.
- But notice: for some x-values (like x=0), there are two y-values?
- Let's check: the graph appears to go up from (-3,-3) to (0,3), then down to (2,-3). But is it symmetric?

Wait — actually, it looks like:
- From (-3, -3) to (0, 3): increasing
- Then from (0, 3) to (2, -3): decreasing
- But at x=0, it's a peak — only one point.

But now look: does it pass vertical line test?

At x=1, for example, there might be two points? No — it's a single continuous curve going up then down.

But wait — the curve may cross itself?

Looking at the graph: it starts at (-3,-3), goes up to (0,3), then down to (2,-3). That’s a parabola-like shape, opening downward.

But if it's a smooth curve, it might still be a function — unless it loops.

But here, it seems like a single U-shaped curve, but inverted — so it's a function if it doesn't go backward.

Wait — the graph appears to go from left to right, but the curve goes up and then down — so it's a function (like a downward parabola).

But check: at x=0, only one point (0,3). At x=1, only one y-value.

But look at the endpoints:
- Left: (-3, -3) — solid dot
- Right: (2, -3) — open circle

And it peaks at (0, 3)

So:
- Domain: x from -3 to 2 → [-3, 2)
- Range: y from -3 to 3 → [-3, 3]
- Function?: Yes — even though it increases then decreases, it’s still a function (each x has one y)

✔ Answer:
- Domain: [-3, 2)
- Range: [-3, 3]
- Function: Yes

Wait — is the right endpoint open? Yes — at (2, -3), open circle, so x=2 not included.

But the graph ends at x=2, open — so domain: [-3, 2)

And y goes from -3 to 3 — both achieved:
- y = -3 at x = -3 and near x=2 (but not at x=2), so y = -3 is achieved at x=-3
- y = 3 at x=0

So range: [-3, 3]

Yes.

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✔ Final Answers:

| # | Domain | Range | Function? |
|---|----------------|----------------|-----------|
| 1 | [0, 2) | [0, 4] | Yes |
| 2 | [-3, 3] | [0, 3] | Yes |
| 3 | (-3, 3] | [-2, 2] | Yes |
| 4 | [-2, 0] | [-2, 2] | No |
| 5 | (-4, 0) | (-1, 3) | Yes |
| 6 | [-3, 3] | [-3, 3] | Yes |
| 7 | (-4, 0] | [-1, 3) | Yes |
| 8 | [-3, 2) | [-3, 3] | Yes |

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✔ Explanation Summary:

- Domain: Look at the leftmost to rightmost x-values.
- Range: Look at the bottom to top y-values.
- Function: Use Vertical Line Test — if any vertical line hits the graph more than once, it's not a function.
- Open circles: Not included in domain/range.
- Closed circles: Included.

Let me know if you'd like this formatted as a printable answer sheet!