Domain and Range Worksheet #1 showing 12 graphs with handwritten answers for domain, range, and function identification.

Let's go through each of the 12 problems on the Domain and Range Worksheet #1 and solve them step by step. For each graph, we will:

1. Determine the domain (all possible x-values).
2. Determine the range (all possible y-values).
3. Determine if it is a function using the Vertical Line Test.
4. If it *is* a function, classify it as discrete, continuous, or neither.

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1) Graph: A U-shaped parabola opening upward, vertex at (-2, -1), passing through (-3,0), (-1,0), etc.

- Domain: All real numbers → $(-\infty, \infty)$
- Range: From the vertex up → $[-1, \infty)$
- Function? YES — passes vertical line test.
- Type: Continuous

✔ Answer:
- Domain: $(-\infty, \infty)$
- Range: $[-1, \infty)$
- Function? YES
- Type: Continuous

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2) Graph: A V-shaped graph (absolute value), vertex at (1, -1), opens upward

- Domain: All real numbers → $(-\infty, \infty)$
- Range: From vertex up → $[-1, \infty)$
- Function? YES — passes vertical line test.
- Type: Continuous

✔ Answer:
- Domain: $(-\infty, \infty)$
- Range: $[-1, \infty)$
- Function? YES
- Type: Continuous

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3) Graph: A wavy curve that goes up and down (like a sine wave), but only over a finite interval

Looking closely: It starts at $x = -2$, ends at $x = 2$, peaks around $y=3$, dips to $y=-1$

- Domain: $[-2, 2]$
- Range: $[-1, 3]$
- Function? YES — no vertical line intersects more than once.
- Type: Continuous

✔ Answer:
- Domain: $[-2, 2]$
- Range: $[-1, 3]$
- Function? YES
- Type: Continuous

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4) Graph: A horizontal line segment from $x = -2$ to $x = 2$, at $y = 2$

- Domain: $[-2, 2]$
- Range: $\{2\}$ or $[2, 2]$ → just $y=2$
- Function? YES — every x has one y.
- Type: Continuous

✔ Answer:
- Domain: $[-2, 2]$
- Range: $\{2\}$
- Function? YES
- Type: Continuous

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5) Graph: Scattered points: (-3,-2), (-2,-1), (-1,0), (0,1), (1,2)

- Domain: $x$-values: $-3, -2, -1, 0, 1$ → $\{-3, -2, -1, 0, 1\}$
- Range: $y$-values: $-2, -1, 0, 1, 2$ → $\{-2, -1, 0, 1, 2\}$
- Function? YES — each x maps to exactly one y.
- Type: Discrete (points not connected)

✔ Answer:
- Domain: $\{-3, -2, -1, 0, 1\}$
- Range: $\{-2, -1, 0, 1, 2\}$
- Function? YES
- Type: Discrete

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6) Graph: A diagonal line from top-left to bottom-right, starting at $(-2,3)$, ending at $(2,-1)$

- Domain: $[-2, 2]$
- Range: $[-1, 3]$
- Function? YES — straight line, passes vertical line test.
- Type: Continuous

✔ Answer:
- Domain: $[-2, 2]$
- Range: $[-1, 3]$
- Function? YES
- Type: Continuous

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7) Graph: A circle centered at origin, radius 2

- Domain: $[-2, 2]$
- Range: $[-2, 2]$
- Function? NO — fails vertical line test (e.g., at $x=0$, two y-values)
- Type: Not applicable (not a function)

✔ Answer:
- Domain: $[-2, 2]$
- Range: $[-2, 2]$
- Function? NO

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8) Graph: A "W" shape, like a cubic with three humps, going from left to right

Looks like a continuous curve with multiple peaks and valleys.

- Domain: All real numbers → $(-\infty, \infty)$
- Range: From lowest point to highest — appears to go from $y=-2$ to $y=2$ → $[-2, 2]$
- Function? YES — passes vertical line test.
- Type: Continuous

✔ Answer:
- Domain: $(-\infty, \infty)$
- Range: $[-2, 2]$
- Function? YES
- Type: Continuous

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9) Graph: A zigzag line connecting several points, with a "V" shape pointing down

But note: it has one x-value corresponding to two different y-values (e.g., at $x=1$, there are two outputs). So it fails vertical line test.

- Domain: $[-3, 3]$
- Range: $[-2, 2]$
- Function? NO — fails vertical line test
- Type: Not applicable

✔ Answer:
- Domain: $[-3, 3]$
- Range: $[-2, 2]$
- Function? NO

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10) Graph: A series of disconnected line segments forming a "staircase" pattern

Points: (-3,-1), (-2,-1), (-1,0), (0,1), (1,2), (2,2)

- Domain: $x$ values: $-3, -2, -1, 0, 1, 2$ → $\{-3, -2, -1, 0, 1, 2\}$
- Range: $y$ values: $-1, 0, 1, 2$ → $\{-1, 0, 1, 2\}$
- Function? YES — each x has one y.
- Type: Discrete (segments are disconnected, not smooth)

✔ Answer:
- Domain: $\{-3, -2, -1, 0, 1, 2\}$
- Range: $\{-1, 0, 1, 2\}$
- Function? YES
- Type: Discrete

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11) Graph: A broken line graph, starting at (-3,-2), going up to (-1,2), then down to (1,1), then to (3,0)

It’s a piecewise linear graph, all connected.

- Domain: $[-3, 3]$
- Range: From $y=-2$ to $y=2$ → $[-2, 2]$
- Function? YES — passes vertical line test.
- Type: Continuous (connected line segments)

✔ Answer:
- Domain: $[-3, 3]$
- Range: $[-2, 2]$
- Function? YES
- Type: Continuous

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12) Graph: A triangle-like shape, but drawn as lines connecting points: from (-2,-1), to (0,2), to (2,-1), back to (-2,-1)? Wait, no — looks like a "V" flipped

Wait — actually, it's a triangle formed by connecting:

- $(-2, -1)$ to $(0, 2)$ to $(2, -1)$, and then back?

But wait — the last segment seems to go from $(2,-1)$ to $(-2,-1)$? That would be a full triangle.

But now, at $x=0$, there are two y-values? No — let’s see.

Actually, the graph shows:

- A line from $(-2, -1)$ to $(0, 2)$
- Then from $(0, 2)$ to $(2, -1)$
- Then from $(2, -1)$ to $(-2, -1)$? But that would make a triangle.

But if you draw it, at $x=1$, only one y-value; at $x=0$, only $y=2$. But what about the base?

Wait — looking carefully: The graph connects $(-2,-1)$ to $(0,2)$, then $(0,2)$ to $(2,-1)$, and then from $(2,-1)$ back to $(-2,-1)$ via a straight line? That would create a triangle.

But in that case, for any $x$ between $-2$ and $2$, there are two y-values (top and bottom), except at endpoints.

So for example, at $x=1$, there is one point on the upper line and one on the lower line → fails vertical line test.

Thus, NOT a function.

- Domain: $[-2, 2]$
- Range: From $y=-1$ to $y=2$ → $[-1, 2]$
- Function? NO — fails vertical line test
- Type: Not applicable

✔ Answer:
- Domain: $[-2, 2]$
- Range: $[-1, 2]$
- Function? NO

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## ✔ Final Answers Summary:

| # | Domain | Range | Function? | Type |
|---|----------------------|--------------------|-----------|--------------|
| 1 | $(-\infty, \infty)$ | $[-1, \infty)$ | YES | Continuous |
| 2 | $(-\infty, \infty)$ | $[-1, \infty)$ | YES | Continuous |
| 3 | $[-2, 2]$ | $[-1, 3]$ | YES | Continuous |
| 4 | $[-2, 2]$ | $\{2\}$ | YES | Continuous |
| 5 | $\{-3, -2, -1, 0, 1\}$ | $\{-2, -1, 0, 1, 2\}$ | YES | Discrete |
| 6 | $[-2, 2]$ | $[-1, 3]$ | YES | Continuous |
| 7 | $[-2, 2]$ | $[-2, 2]$ | NO | — |
| 8 | $(-\infty, \infty)$ | $[-2, 2]$ | YES | Continuous |
| 9 | $[-3, 3]$ | $[-2, 2]$ | NO | — |
|10 | $\{-3, -2, -1, 0, 1, 2\}$ | $\{-1, 0, 1, 2\}$ | YES | Discrete |
|11 | $[-3, 3]$ | $[-2, 2]$ | YES | Continuous |
|12 | $[-2, 2]$ | $[-1, 2]$ | NO | — |

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📌 Key Concepts Recap:

- Domain: All x-values where the graph exists.
- Range: All y-values where the graph exists.
- Function? Use Vertical Line Test: If any vertical line crosses the graph more than once → NOT a function.
- Discrete: Points not connected (like dots).
- Continuous: Connected without breaks (curves, lines).

Let me know if you'd like this in printable format or need help with similar problems!