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).

---

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

---

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

---

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

---

4) Graph 4

- Graph Description: A curved shape like a sideways "U" or "C", from (-3, 0) to (0, 3), then back down to (0, -3)? Wait — actually, looks like two parts:
- From (-3, 0) to (0, 3): increasing
- From (0, -3) to (0, 0): decreasing?
But wait — it appears to have two y-values for x = 0, so it fails vertical line test.
Actually, the graph shows a curve from (-3, 0) to (0, 3) with solid dot at (-3, 0), open at (0, 3), and another curve from (0, -3) to (0, 0) with open at (0, -3), solid at (0, 0)? No.

Wait — let's re-analyze carefully.

Looking at graph 4: It’s a semi-circular shape opening to the right, centered at origin, radius 3. Starts at (-3, 0), goes up to (0, 3), then down to (0, -3), but ends at (0, -3) with open circle?

Wait — actually, the graph seems to be a semicircle to the left, from (-3, 0) to (0, 3) and (0, -3), but it's not connected at (0, 0). There are two separate curves?

No — better interpretation: This is a circle-like shape that starts at (-3, 0), goes up to (0, 3), then down to (0, -3), but only includes the left half of the circle (x ≤ 0). However, there are two points at x = 0 — one at (0, 3) and (0, -3)? But they are not both filled.

Actually, looking closely:
- The top curve: from (-3, 0) to (0, 3), solid at (-3, 0), open at (0, 3)
- The bottom curve: from (0, -3) to (-3, 0), solid at (-3, 0), open at (0, -3)

So this is a closed loop? No — it's a single continuous curve forming a semicircle on the left side of the origin.

But wait — at x = -2, for example, there are two y-values — one positive and one negative. So it's not a function.

Yes! Vertical line test fails — e.g., draw x = -2, hits two points.

So:
- Domain: x from -3 to 0 → [-3, 0]
- Range: y from -3 to 3 → [-3, 3]
- Function?: No — fails vertical line test.
- ✔ Answer:
- Domain: [-3, 0]
- Range: [-3, 3]
- Function: No

---

5) Graph 5

- Graph Description: Two lines forming a "V" shape, but flipped upside-down. Starts at (-3, 3) with open circle, goes down to (0, 0), then up to (2, 0) with solid dot, and continues down to (3, -2) with open circle.
- Wait — actually, it looks like:
- From (-3, 3) to (0, 0): solid line, open at (-3, 3)
- Then from (0, 0) to (3, -2): solid line, open at (3, -2)
But at (0, 0), it's a corner — solid dot.
Also, at (2, 0), there's a point — but it's just on the line.

Wait — no, it appears to be one piece: from (-3, 3) to (0, 0), then from (0, 0) to (3, -2). But at (0, 0), it's a vertex.

But the key: is it a function?

Yes — each x has one y.

- Domain: x from -3 to 3 → (-3, 3)
- Range: y from -2 to 3 → [-2, 3)
- At x = -3, open circle → y = 3 not included
- At x = 3, open circle → y = -2 not included? Wait — but y = -2 is at x = 3, which is open → so y = -2 not included?
But the line goes to (3, -2), open circle → so y approaches -2 but doesn't reach it.

But look: the lowest point is at (3, -2), open circle → so y > -2? No — it's decreasing toward -2.

But the graph goes from (-3, 3) to (3, -2), so y decreases from 3 to -2.

But since both ends are open, y does not include 3 or -2.

So range: (-2, 3)

Wait — but at x=0, y=0, which is included.

So:
- Domain: (-3, 3)
- Range: (-2, 3)
- Function?: Yes — one y per x
- ✔ Answer:
- Domain: (-3, 3)
- Range: (-2, 3)
- Function: Yes

Wait — but the line goes from (-3, 3) to (0, 0), then from (0, 0) to (3, -2). So it's continuous.

But at x = -3, open → x > -3
At x = 3, open → x < 3

So domain: (-3, 3)

y values: from just below 3 down to just above -2 → so range: (-2, 3)

Yes.

---

6) Graph 6

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

---

7) Graph 7

- Graph Description: A straight line from (-3, 2) to (1, -2), with open circles at both ends.
- Domain: x from -3 to 1 → (-3, 1)
- Range: y from -2 to 2 → (-2, 2)
- Since both ends are open, neither endpoint is included.
- Function?: Yes — straight line, one output per input.
- ✔ Answer:
- Domain: (-3, 1)
- Range: (-2, 2)
- Function: Yes

---

8) Graph 8

- Graph Description: A curved shape — starts at (-3, 2) with open circle, goes down to (0, -3), then up to (3, 2) with open circle. It's symmetric — like a "U" shape opening upward.
- But wait — at x = 0, y = -3, solid dot? Or open?

Looking closely:
- From (-3, 2) to (0, -3): solid line, open at (-3, 2), solid at (0, -3)
- From (0, -3) to (3, 2): solid line, open at (3, 2), solid at (0, -3)

So it's a parabola-like shape, vertex at (0, -3), opens upward.

But now check vertical line test: at x = 1, there is only one y-value — yes.

Is it a function? Yes — because it's a U-shaped curve, but only one y per x.

Wait — but look: the graph is continuous and symmetric, but each x has exactly one y.

So it is a function.

- Domain: x from -3 to 3 → (-3, 3)
- Both ends open → so x ≠ -3 and x ≠ 3
- Range: y from -3 to 2 → [-3, 2)
- At x = 0, y = -3, solid dot → included
- At x = ±3, y = 2, but open circles → y = 2 not included
- So range: [-3, 2)
- Function?: Yes
- ✔ Answer:
- Domain: (-3, 3)
- Range: [-3, 2)
- Function: Yes

---

✔ Final Answers:

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

---

🔍 Explanation Summary:

- Domain: Look at the horizontal span of the graph.
- Range: Look at the vertical span.
- Function: Use the Vertical Line Test — if any vertical line crosses the graph more than once, it's not a function.
- Open circles mean the endpoint is not included (use parentheses).
- Closed circles mean the endpoint is included (use brackets).

Let me know if you'd like a visual explanation or printable version!