Domain And Range Of Graphs Worksheet Answers - Printable Template ... - Free Printable

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, with open circle at endpoint

Line from $(-3, 3)$ to $(2, -2)$, but open circle at $(-3,3)$, so not included.

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

✔ Answer:
- Domain: $(-3, 2]$
- Range: $[-2, 3)$
- 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: Neither (not a function)

✔ Answer:
- Domain: $[-2, 2]$
- Range: $[-2, 2]$
- Function? NO
- Type: Neither

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8) Graph: A periodic wave-like graph, seems like a cosine or sine wave over $(-\infty, \infty)$

Looks like it oscillates between $y = -2$ and $y = 2$, extends infinitely in both directions.

- Domain: $(-\infty, \infty)$
- Range: $[-2, 2]$
- 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 with multiple branches going from same x-value to different y-values

This graph has one x-value mapping to multiple y-values, e.g., at $x = 0$, there are two outputs.

- Domain: $[-2, 2]$
- Range: $[-2, 2]$
- Function? NO — fails vertical line test.
- Type: Neither

✔ Answer:
- Domain: $[-2, 2]$
- Range: $[-2, 2]$
- Function? NO
- Type: Neither

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10) Graph: A series of disconnected line segments forming a "zigzag" shape

Points: $(-3,-2)$, $(-2,-1)$, $(-1,0)$, $(0,1)$, $(1,2)$, $(2,1)$, $(3,0)$ — but not all connected?

Wait — actually, it’s a piecewise linear graph, connected segments.

From left to right:
- From $(-3,-2)$ to $(-2,-1)$
- Then jumps to $(-1,0)$ to $(0,1)$
- Then to $(1,2)$ to $(2,1)$ to $(3,0)$

But wait — there's a gap between $x = -2$ and $x = -1$. So not continuous.

But each x has one y → function?

Yes — even with gaps, as long as each x maps to one y.

- Domain: $[-3, -2] \cup [-1, 3]$
- Range: $[-2, 2]$
- Function? YES — passes vertical line test.
- Type: Neither discrete nor continuous — it's piecewise but not fully connected → neither? Wait…

Actually, since it's made of connected line segments, but with a gap, it's not continuous, but still a function.

But “neither” usually means not discrete and not continuous. Since it’s not discrete (it has lines), and not continuous (due to gap), then neither is correct.

But some curricula say: if it's not continuous, but not discrete (i.e., not isolated points), then it's neither.

✔ Answer:
- Domain: $[-3, -2] \cup [-1, 3]$
- Range: $[-2, 2]$
- Function? YES
- Type: Neither

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11) Graph: A V-shaped graph with a break at $x = 0$, but still one output per x

Looks like two rays: from $(-2, -2)$ to $(0, 0)$, then from $(0, 0)$ to $(2, -2)$ — but wait, does it have a jump?

No — it looks like a V-shape opening downward, starting at $(-2,-2)$, rising to $(0,0)$, then dropping to $(2,-2)$ — continuous.

Wait — but look carefully: Is it a single continuous path?

Yes — it's a V shape, symmetric, continuous.

- Domain: $[-2, 2]$
- Range: $[-2, 0]$
- Function? YES — each x has one y.
- Type: Continuous

✔ Answer:
- Domain: $[-2, 2]$
- Range: $[-2, 0]$
- Function? YES
- Type: Continuous

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12) Graph: A triangle-shaped graph with a peak at (0,2), and two sides going down to (-2,0) and (2,0)

Wait — but the graph has two outputs for some x-values?

No — actually, it's a triangle with vertices at $(-2,0)$, $(0,2)$, $(2,0)$ — connected.

So it's a V-shaped upside-down V — yes, a continuous function.

But wait — the student wrote NO for function — but that's wrong!

Wait — let’s check: Is it a function?

At $x=0$, only one point: $y=2$

At any other $x$, only one $y$

And it’s a continuous line from $(-2,0)$ to $(0,2)$ to $(2,0)$

So it is a function.

But the student wrote NO — this is incorrect.

Wait — perhaps the graph shows something else?

Wait — look again: The graph might be two separate lines going from the same x to two different y-values?

No — the image shows a triangle with a peak at top, so it's a single continuous curve.

So it should be a function.

But the student marked NO — likely a mistake.

But let's double-check: Does any vertical line intersect it twice?

Only at the apex? No — at $x=0$, only one point.

Elsewhere, only one intersection.

So YES, it is a function.

Wait — unless the graph is not a single curve, but two lines from same x?

But based on typical worksheets, this is a V-shaped graph, which is a function.

But the student wrote NO — probably an error.

But let’s suppose the graph is not a function — maybe it's a diamond or something?

Wait — the graph is drawn as a triangle with three vertices: $(-2,0)$, $(0,2)$, $(2,0)$ — connected.

That’s a function — it's $f(x) = 2 - |x|$ for $|x| \leq 2$

So it is a function.

✔ Correct answer:
- Domain: $[-2, 2]$
- Range: $[0, 2]$
- Function? YES
- Type: Continuous

But the student wrote NO — this is incorrect.

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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 | $(-3, 2]$ | $[-2, 3)$ | YES | Continuous |
| 7 | $[-2, 2]$ | $[-2, 2]$ | NO | Neither |
| 8 | $(-\infty, \infty)$ | $[-2, 2]$ | YES | Continuous |
| 9 | $[-2, 2]$ | $[-2, 2]$ | NO | Neither |
|10 | $[-3, -2] \cup [-1, 3]$ | $[-2, 2]$ | YES | Neither |
|11 | $[-2, 2]$ | $[-2, 0]$ | YES | Continuous |
|12 | $[-2, 2]$ | $[0, 2]$ | YES | Continuous |

> ❗️Note: Student incorrectly answered NO for #12 — it is a function.

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🔍 Explanation of Key Concepts:

- Domain: All x-values covered by the graph.
- Range: All y-values covered.
- Function?: Use Vertical Line Test — if any vertical line crosses the graph more than once → not a function.
- Discrete: Points are isolated (no connections).
- Continuous: Connected smooth curve/line (no breaks).
- Neither: Not discrete (has connections), but not continuous (has breaks or jumps).

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✔ Final Note:

The student made a mistake on #12 — it is a function. Also, #10 may be ambiguous depending on whether it's connected or not, but if it's not connected (with a gap), it's neither discrete nor continuous.

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