Domain and Range Worksheet #1 featuring 12 graphs to analyze for domain, range, and function status.

Thank you for uploading the image. I’ll solve each of the 12 problems on “Domain and Range Worksheet #1”, explaining the concepts as I go.

---

Key Concepts:

- Domain: All possible x-values (inputs) for which the graph exists.
- Range: All possible y-values (outputs) that the graph reaches.
- Function?: A relation is a function if every x-value has only one y-value (passes the Vertical Line Test — no vertical line intersects the graph more than once).
- Discrete vs Continuous:
- Discrete: Graph consists of separate, distinct points (often from data sets or integers).
- Continuous: Graph is an unbroken curve or line (can take any value in an interval).

---

Let’s go problem by problem:

---

1)

Graph: A parabola opening upward, vertex at (0, -1), extends infinitely left and right.

✔ Domain: All real numbers → `(-∞, ∞)`
✔ Range: Starts at y = -1 and goes up → `[-1, ∞)`
✔ Function?: YES — passes vertical line test.
✔ Type: Continuous

---

2)

Graph: A piecewise graph with sharp turns, appears to be defined from x = -5 to x = 5, with endpoints included.

✔ Domain: `[-5, 5]`
✔ Range: From y = -3 to y = 4 → `[-3, 4]`
✔ Function?: YES — every x has only one y.
✔ Type: Continuous (connected lines)

---

3)

Graph: Wavy curve, looks like a sine wave, continuous and smooth.

✔ Domain: `(-∞, ∞)`
✔ Range: Appears to oscillate between y = -2 and y = 2 → `[-2, 2]`
✔ Function?: YES — passes vertical line test.
✔ Type: Continuous

---

4)

Graph: Two horizontal lines — top at y=3 (from x=-4 to x=2), bottom at y=-2 (from x=-4 to x=2). But notice: at x=-4, x=0, x=2, there are open and closed circles indicating discrete points.

Actually, looking closely — this graph shows only specific x-values with points plotted (likely discrete). For example, at x = -4, -2, 0, 2 — there are points. Not a continuous line.

✔ Domain: {-4, -2, 0, 2} — only those x-values have points.
✔ Range: {-2, 3} — only those y-values appear.
✔ Function?: YES — each x has only one y.
✔ Type: Discrete

*(Note: If it were continuous lines, domain/range would be intervals, but here it's clearly plotted points.)*

---

5)

Graph: Looks like a “W” shape — continuous curve.

✔ Domain: `(-∞, ∞)`
✔ Range: Minimum y ≈ -1, maximum y ≈ 3 → `[-1, 3]`
✔ Function?: YES — passes vertical line test.
✔ Type: Continuous

---

6)

Graph: Downward-opening parabola, vertex at (0, 3), opens down.

✔ Domain: `(-∞, ∞)`
✔ Range: y ≤ 3 → `(-∞, 3]`
✔ Function?: YES
✔ Type: Continuous

---

7)

Graph: A circle centered at origin, radius 2.

✔ Domain: x from -2 to 2 → `[-2, 2]`
✔ Range: y from -2 to 2 → `[-2, 2]`
✔ Function?: NO — fails vertical line test (e.g., at x=0, y=±2).
✔ Type: Continuous (but not a function)

---

8)

Graph: “M” or “W” shaped — continuous, symmetric.

✔ Domain: `(-∞, ∞)`
✔ Range: From y = -1 to y = 3 → `[-1, 3]`
✔ Function?: YES — passes vertical line test.
✔ Type: Continuous

---

9)

Graph: A downward curve starting high on left, decreasing to right — looks like part of a parabola or exponential decay.

✔ Domain: `(-∞, ∞)`
✔ Range: y < 4 (never reaches 4, open at top) → `(-∞, 4)`
✔ Function?: YES
✔ Type: Continuous

---

10)

Graph: Zigzag line — starts at (-4, 2), goes down to (-2, -2), up to (0, 2), down to (2, -2), up to (4, 2). Connected lines.

✔ Domain: `[-4, 4]`
✔ Range: `[-2, 2]`
✔ Function?: YES — each x has one y.
✔ Type: Continuous

---

11)

Graph: Two line segments — one from (-4, -2) to (0, 2), another from (0, 2) to (4, -2). Continuous at (0,2).

✔ Domain: `[-4, 4]`
✔ Range: `[-2, 2]`
✔ Function?: YES — passes vertical line test.
✔ Type: Continuous

---

12)

Graph: Two lines forming a “V” shape, but both lines start at the same point (0, 0) and go outward — at x=0, there are two y-values? Wait — actually, both lines pass through (0,0) and go up/down? Let’s look again.

Actually — this graph shows two lines crossing at (0,0) — one going up-right, one going down-right. So at x > 0, two different y-values? NO — wait, actually, it’s likely two rays: one from (0,0) up to (4,4), and one from (0,0) down to (4,-4)?

If so — then for x > 0, there are two y-values → NOT a function.

✔ Domain: `[0, 4]` (assuming graph stops at x=4)
✔ Range: `[-4, 4]`
✔ Function?: NO — fails vertical line test (e.g., at x=2, y=2 and y=-2)
✔ Type: Continuous (but not a function)

*(Note: This is often called a “relation” or “graph of a relation” — not a function.)*

---

## ✔ Final Answers Summary:

| # | Domain | Range | Function? | Type |
|---|------------------|------------------|----------|------------|
| 1 | (-∞, ∞) | [-1, ∞) | Yes | Continuous |
| 2 | [-5, 5] | [-3, 4] | Yes | Continuous |
| 3 | (-∞, ∞) | [-2, 2] | Yes | Continuous |
| 4 | {-4, -2, 0, 2} | {-2, 3} | Yes | Discrete |
| 5 | (-∞, ∞) | [-1, 3] | Yes | Continuous |
| 6 | (-∞, ∞) | (-∞, 3] | Yes | Continuous |
| 7 | [-2, 2] | [-2, 2] | No | Continuous |
| 8 | (-∞, ∞) | [-1, 3] | Yes | Continuous |
| 9 | (-∞, ∞) | (-∞, 4) | Yes | Continuous |
|10 | [-4, 4] | [-2, 2] | Yes | Continuous |
|11 | [-4, 4] | [-2, 2] | Yes | Continuous |
|12 | [0, 4] | [-4, 4] | No | Continuous |

---

📌 Important Note for #4 & #12:
- #4 is discrete because only specific points are plotted (not connected lines).
- #12 is not a function because vertical lines intersect it twice for x > 0.

---

✔ Your name “Fatima Saleh” is written — great job working through this! Let me know if you’d like me to explain any one problem in even more detail.