Domain and Range Worksheet #1 featuring 12 graphs for analyzing functions.

Let's go through each graph on the worksheet and determine:

1. Domain – all possible x-values (horizontal extent of the graph)
2. Range – all possible y-values (vertical extent of the graph)
3. Function? – Use the Vertical Line Test: If any vertical line intersects the graph more than once, it’s not a function.
4. Discrete, Continuous, or Neither?
- Discrete: Points are separate and not connected (like dots).
- Continuous: A smooth unbroken curve or line.
- Neither: Some mix or broken pattern.

We’ll analyze each graph one by one.

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1) Graph: Vertical line at x = 0

- Domain: Only x = 0 → $ \{0\} $
- Range: From y = -3 to y = 3 → $ [-3, 3] $
- Function? Yes — only one output per input (only one point for x=0)
- Type: Discrete (single vertical line with points)

✔ Answer:
Domain: $ \{0\} $
Range: $ [-3, 3] $
Function? Yes
Type: Discrete

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2) Graph: Zig-zag line from (-5,-5) to (5,5), with sharp turns

- Domain: From x = -5 to x = 5 → $ [-5, 5] $
- Range: From y = -5 to y = 5 → $ [-5, 5] $
- Function? Yes — passes vertical line test (no vertical line hits more than once)
- Type: Continuous (connected lines)

✔ Answer:
Domain: $ [-5, 5] $
Range: $ [-5, 5] $
Function? Yes
Type: Continuous

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3) Graph: Wave-like curve (sinusoidal shape)

- Domain: From x = -3 to x = 3 → $ [-3, 3] $
- Range: From y = -2 to y = 2 → $ [-2, 2] $
- Function? Yes — every x has one y
- Type: Continuous

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

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4) Graph: Horizontal line segment from (-2, 2) to (2, 2)

- Domain: x goes from -2 to 2 → $ [-2, 2] $
- Range: y is always 2 → $ \{2\} $
- Function? Yes — one y-value for all x
- Type: Continuous (line segment)

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

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5) Graph: Two disconnected horizontal segments

- Top segment: from (-2, 3) to (2, 3)
- Bottom segment: from (-2, -3) to (2, -3)
- So two horizontal lines at y = 3 and y = -3, both from x = -2 to x = 2
- Domain: $ [-2, 2] $
- Range: $ \{-3, 3\} $
- Function? No — for example, at x = 0, there are two outputs: y = 3 and y = -3 → fails vertical line test
- Type: Discrete (two separate lines, but still connected horizontally)

Wait: The lines are continuous in x, but multiple y-values → not a function.

✔ Answer:
Domain: $ [-2, 2] $
Range: $ \{-3, 3\} $
Function? No
Type: Discrete (since it's made of distinct horizontal lines, though they’re continuous within themselves)

> Note: "Discrete" here refers to the set of points being disconnected vertically. Even if each piece is continuous, the overall relation is not a function.

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6) Graph: Curve starting at (-4, 4), going down to (0, 0), then continuing to (4, -4)

Looks like a downward-opening parabola or a cubic?

Actually, it looks like a cubic-like or parabola, but let's see:
- Starts at (-4, 4), goes down through origin, ends at (4, -4)
- Smooth curve

- Domain: $ [-4, 4] $
- Range: $ [-4, 4] $
- Function? Yes — passes vertical line test
- Type: Continuous

✔ Answer:
Domain: $ [-4, 4] $
Range: $ [-4, 4] $
Function? Yes
Type: Continuous

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7) Graph: Circle centered at origin, radius 2

- Equation: $ x^2 + y^2 = 4 $
- Domain: $ [-2, 2] $
- Range: $ [-2, 2] $
- Function? No — vertical line at x=0 hits two points (top and bottom)
- Type: Continuous (but not a function)

✔ Answer:
Domain: $ [-2, 2] $
Range: $ [-2, 2] $
Function? No
Type: Continuous

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8) Graph: W-shaped curve (like a cosine wave)

- Looks like a quadratic-like or cosine wave, symmetric, peaks at top and bottom
- From x = -4 to x = 4
- Peaks at y = 3, dips to y = -3
- Domain: $ [-4, 4] $
- Range: $ [-3, 3] $
- Function? Yes — no vertical line hits twice
- Type: Continuous

✔ Answer:
Domain: $ [-4, 4] $
Range: $ [-3, 3] $
Function? Yes
Type: Continuous

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9) Graph: Diagonal lines forming an “X” shape

- Lines from (-3, 3) to (3, -3) and from (-3, -3) to (3, 3)
- Two diagonal lines crossing at origin
- Domain: $ [-3, 3] $
- Range: $ [-3, 3] $
- Function? No — at x=0, two outputs: y=0 and y=0? Wait — actually, they cross at (0,0), but for x ≠ 0, e.g., x=1, there are two y-values: one on each line
- For x=1: y=1 and y=-1 → fails vertical line test
- Type: Continuous (lines are continuous), but not a function

✔ Answer:
Domain: $ [-3, 3] $
Range: $ [-3, 3] $
Function? No
Type: Continuous

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10) Graph: V-shaped graph (absolute value), vertex at (0,0), goes up to (2,4) and (-2,4)

- Looks like $ y = |x| \times 2 $? From (-2,4) to (0,0) to (2,4)
- Domain: $ [-2, 2] $
- Range: $ [0, 4] $
- Function? Yes — one y per x
- Type: Continuous

✔ Answer:
Domain: $ [-2, 2] $
Range: $ [0, 4] $
Function? Yes
Type: Continuous

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11) Graph: Piecewise linear graph

- Starts at (-2, 0), goes up to (0, 2), then down to (2, 0)
- Then another segment from (2, 0) to (4, -2), then up to (6, 0)
- So two "V" shapes?
- Actually, it looks like a zigzag:
- From (-2,0) to (0,2): up
- (0,2) to (2,0): down
- (2,0) to (4,-2): down
- (4,-2) to (6,0): up

So domain: x from -2 to 6 → $ [-2, 6] $
Range: y from -2 to 2 → $ [-2, 2] $

Passes vertical line test → Function? Yes
Lines are connected → Continuous

✔ Answer:
Domain: $ [-2, 6] $
Range: $ [-2, 2] $
Function? Yes
Type: Continuous

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12) Graph: Two lines forming an “X” between (-2,2) and (2,-2)

- One line from (-2,2) to (2,-2)
- Another from (-2,-2) to (2,2)
- Cross at (0,0)

This is similar to #9 — same idea

- Domain: $ [-2, 2] $
- Range: $ [-2, 2] $
- Function? No — vertical line at x=1 hits two points: (1,1) and (1,-1)
- Type: Continuous

✔ Answer:
Domain: $ [-2, 2] $
Range: $ [-2, 2] $
Function? No
Type: Continuous

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✔ Final Answers Summary:

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

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