Domain and Range Worksheet #1 | Exercises Calculus | Docsity - Free Printable

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

- State the domain (all possible x-values)
- State the range (all possible y-values)
- Determine if it is a function (using the vertical line test: if any vertical line intersects the graph more than once, it's not a function)
- If it is a function, classify it as discrete, continuous, or neither

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1) Graph: Two vertical lines at x = -3 and x = 2

- Domain: $ x = -3 $ and $ x = 2 $ → $ \{-3, 2\} $
- Range: All real numbers from $-\infty$ to $\infty$, but only values where points exist. The graph shows arrows going up and down vertically, so all real numbers → $ (-\infty, \infty) $
- Function? No — because for $x = -3$ and $x = 2$, there are multiple y-values (the entire vertical line), violating the definition of a function.
- Answer:
Domain: $ \{-3, 2\} $
Range: $ (-\infty, \infty) $
Function? No

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2) Graph: Zig-zag line from (-4, -1) to (6, 0) with open and closed circles

- Domain: From $x = -4$ to $x = 6$. Open circle at $x = -4$, closed at $x = 6$ → $ [-4, 6] $
- Range: From $y = -1$ to $y = 3$ → $ [-1, 3] $
- Function? Yes — passes vertical line test.
- Type: Continuous (connected lines)
- Answer:
Domain: $ [-4, 6] $
Range: $ [-1, 3] $
Function? Yes (Continuous)

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3) Graph: Cubic-like curve with turning points

- Domain: All real numbers → $ (-\infty, \infty) $
- Range: All real numbers → $ (-\infty, \infty) $
- Function? Yes — passes vertical line test.
- Type: Continuous
- Answer:
Domain: $ (-\infty, \infty) $
Range: $ (-\infty, \infty) $
Function? Yes (Continuous)

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4) Graph: Horizontal line at y = 2, extending infinitely left and right

- Domain: All real numbers → $ (-\infty, \infty) $
- Range: Only $y = 2$ → $ \{2\} $
- Function? Yes — every x maps to one y-value.
- Type: Continuous
- Answer:
Domain: $ (-\infty, \infty) $
Range: $ \{2\} $
Function? Yes (Continuous)

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5) Graph: Scattered points

- Points:
- $(-3, 1)$, $(-2, 1)$, $(-1, 1)$, $(0, 4)$, $(1, 1)$, $(2, 1)$, $(3, 1)$, $(4, 1)$, $(5, 1)$, and $(1, -2)$
- Wait — at $x=1$, two points: $y=1$ and $y=-2$
- So two outputs for $x=1$ → Not a function
- Domain: $ x = -3, -2, -1, 0, 1, 2, 3, 4, 5 $ → $ \{-3, -2, -1, 0, 1, 2, 3, 4, 5\} $
- Range: $ y = -2, 1, 4 $ → $ \{-2, 1, 4\} $
- Function? No
- Answer:
Domain: $ \{-3, -2, -1, 0, 1, 2, 3, 4, 5\} $
Range: $ \{-2, 1, 4\} $
Function? No

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6) Graph: Parabola opening downward, vertex at (0, 2), ends at (-3, 0) and (3, 0)

- Domain: From $x = -3$ to $x = 3$ → $ [-3, 3] $
- Range: From $y = 0$ to $y = 2$ → $ [0, 2] $
- Function? Yes — passes vertical line test.
- Type: Continuous
- Answer:
Domain: $ [-3, 3] $
Range: $ [0, 2] $
Function? Yes (Continuous)

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7) Graph: Circle centered at (0, 0), radius 2

- Domain: $ x \in [-2, 2] $
- Range: $ y \in [-2, 2] $
- Function? No — fails vertical line test (e.g., at $x=0$, two y-values: $y=2$ and $y=-2$)
- Answer:
Domain: $ [-2, 2] $
Range: $ [-2, 2] $
Function? No

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8) Graph: Wavy curve resembling sine wave, continuous

- Domain: All real numbers → $ (-\infty, \infty) $
- Range: 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 (Continuous)

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9) Graph: Two line segments forming a "V" shape with open circles

- Left segment: From $(-3, 0)$ to $(-1, 1)$, then down to $(-1, -3)$
- Right segment: From $(-1, 1)$ to $(3, 5)$, but open circle at $x=3$
- Wait — at $x = -1$, two points: one at $y=1$ and one at $y=-3$ → not a function
- Also, the point at $x = -1$ has two outputs → fails vertical line test
- Domain: $ x \in [-3, 3) $
- Range: $ y \in [-3, 5) $
- Function? No
- Answer:
Domain: $ [-3, 3) $
Range: $ [-3, 5) $
Function? No

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10) Graph: V-shaped piecewise function, similar to absolute value

- Shape: Downward V from $x = -4$ to $x = 0$, then upward V from $x = 0$ to $x = 4$
- Domain: $ [-4, 4] $
- Range: Minimum at $y = -3$, maximum at $y = 0$ → $ [-3, 0] $
- Function? Yes — passes vertical line test.
- Type: Continuous
- Answer:
Domain: $ [-4, 4] $
Range: $ [-3, 0] $
Function? Yes (Continuous)

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11) Graph: Broken line segments with open and closed circles

- Segments:
- Left: from $(-4, -3)$ to $(-2, 1)$, open at $(-4, -3)$, closed at $(-2, 1)$
- Then jumps to $(1, 1)$, open at $(1, 1)$, goes to $(2, 0)$, then to $(4, -2)$
- At $x = 1$: There’s an open circle at $y=1$, so no point there.
- But wait: Is there a gap between $x = -2$ and $x = 1$? Yes — so discrete?
- Actually, it's piecewise continuous but with a break.
- However, since it's connected in segments, it's continuous within each piece.
- But overall, it's not continuous due to jump discontinuity.
- But is it a function?
- Check: Each x has only one y-value → yes
- Vertical line test passed.
- Domain: $ [-4, 4] $ excluding $x = 1$? Wait — open circle at $x=1$, but no point at $x=1$, so domain excludes $x=1$?
- But look: from $x = -2$ to $x = 1$: open at $x=1$, so $x=1$ not included.
- Then from $x=1$ to $x=4$: starts at $x=1$ with open circle, so again $x=1$ not included.
- So no point at $x=1$ — domain is $ [-4, -2] \cup (1, 4] $
- Range: From $y = -3$ to $y = 1$ → $ [-3, 1] $
- Function? Yes
- Type: Neither discrete nor continuous (has gaps and disconnected parts) → Neither
- Answer:
Domain: $ [-4, -2] \cup (1, 4] $
Range: $ [-3, 1] $
Function? Yes (Neither)

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12) Graph: Triangle shape, connecting points

- Vertices: $(-2, 0)$, $(0, 4)$, $(2, 0)$, back to $(-2, 0)$
- But it's a triangle — so two lines from $x = -2$ to $x = 0$, and $x = 0$ to $x = 2$, and back?
- Wait — actually, it looks like three sides forming a triangle, but at $x = 0$, two different y-values? Let's see:
- Line from $(-2, 0)$ to $(0, 4)$
- Line from $(0, 4)$ to $(2, 0)$
- Line from $(2, 0)$ to $(-2, 0)$? No — that would be a horizontal base.
- But in the graph, it appears the triangle connects $(-2, 0)$ to $(0, 4)$, $(0, 4)$ to $(2, 0)$, and $(2, 0)$ to $(-2, 0)$ — so bottom edge is from $(-2,0)$ to $(2,0)$
- But then at $x = 0$, there are two points: $y = 4$ and $y = 0$ — so fails vertical line test
- Also, the bottom edge is at $y = 0$, and top at $y = 4$ — so at $x = 0$, both $y = 0$ and $y = 4$ → not a function
- Domain: $ [-2, 2] $
- Range: $ [0, 4] $
- Function? No
- Answer:
Domain: $ [-2, 2] $
Range: $ [0, 4] $
Function? No

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✔ Final Answers Summary:

| # | Domain | Range | Function? | Type |
|---|--------|-------|-----------|------|
| 1 | $ \{-3, 2\} $ | $ (-\infty, \infty) $ | No | — |
| 2 | $ [-4, 6] $ | $ [-1, 3] $ | Yes | Continuous |
| 3 | $ (-\infty, \infty) $ | $ (-\infty, \infty) $ | Yes | Continuous |
| 4 | $ (-\infty, \infty) $ | $ \{2\} $ | Yes | Continuous |
| 5 | $ \{-3, -2, -1, 0, 1, 2, 3, 4, 5\} $ | $ \{-2, 1, 4\} $ | No | — |
| 6 | $ [-3, 3] $ | $ [0, 2] $ | Yes | Continuous |
| 7 | $ [-2, 2] $ | $ [-2, 2] $ | No | — |
| 8 | $ (-\infty, \infty) $ | $ [-2, 2] $ | Yes | Continuous |
| 9 | $ [-3, 3) $ | $ [-3, 5) $ | No | — |
|10 | $ [-4, 4] $ | $ [-3, 0] $ | Yes | Continuous |
|11 | $ [-4, -2] \cup (1, 4] $ | $ [-3, 1] $ | Yes | Neither |
|12 | $ [-2, 2] $ | $ [0, 4] $ | No | — |

> Note: For problems where the graph is a function, you should specify whether it's discrete, continuous, or neither. Discrete means isolated points; continuous means unbroken curves; neither means some combination (like broken lines with gaps).

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