How to find the domain and range of a graph - Quora - Free Printable

Let's go through each graph one by one and determine:

- Domain: The set of all possible input values (x-values).
- Range: The set of all possible output values (y-values).
- Function?: Whether the graph passes the vertical line test — if any vertical line intersects the graph more than once, it’s not a function.

We’ll use interval notation or set notation as appropriate.

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1. $ y = x + 1 $

- Graph: A straight line with slope 1, passing through origin.
- Domain: All real numbers → $ (-\infty, \infty) $
- Range: All real numbers → $ (-\infty, \infty) $
- Function? Yes (every x has exactly one y)

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ (-\infty, \infty) $
- Function? Yes

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2. $ y = -2x - 3 $

- Graph: Straight line with negative slope.
- Domain: All real numbers → $ (-\infty, \infty) $
- Range: All real numbers → $ (-\infty, \infty) $
- Function? Yes

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ (-\infty, \infty) $
- Function? Yes

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3. $ y = x^2 - 5 $

- Graph: Parabola opening upward, vertex at $ (0, -5) $
- Domain: All real numbers → $ (-\infty, \infty) $
- Range: From minimum value $ y = -5 $ upward → $ [-5, \infty) $
- Function? Yes (passes vertical line test)

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ [-5, \infty) $
- Function? Yes

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4. $ y = |x| + 3 $

- Graph: V-shaped graph, vertex at $ (0, 3) $
- Domain: All real numbers → $ (-\infty, \infty) $
- Range: Minimum at $ y = 3 $ → $ [3, \infty) $
- Function? Yes

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ [3, \infty) $
- Function? Yes

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5. $ y = -x^2 + 3 $

- Graph: Parabola opening downward, vertex at $ (0, 3) $
- Domain: All real numbers → $ (-\infty, \infty) $
- Range: Max at $ y = 3 $, goes down to $ -\infty $ → $ (-\infty, 3] $
- Function? Yes

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ (-\infty, 3] $
- Function? Yes

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6. $ y = |x| - 4 $

- Graph: V-shaped, vertex at $ (0, -4) $
- Domain: All real numbers → $ (-\infty, \infty) $
- Range: Minimum at $ y = -4 $ → $ [-4, \infty) $
- Function? Yes

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ [-4, \infty) $
- Function? Yes

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7. $ x = -5 $

- Graph: Vertical line at $ x = -5 $
- Domain: Only $ x = -5 $ → $ \{-5\} $
- Range: All real y-values → $ (-\infty, \infty) $
- Function? No — fails vertical line test (one x-value corresponds to infinitely many y-values)

✘ Answer:
- Domain: $ \{-5\} $
- Range: $ (-\infty, \infty) $
- Function? No

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8. $ y = 4 $

- Graph: Horizontal line at $ y = 4 $
- Domain: All real x-values → $ (-\infty, \infty) $
- Range: Only $ y = 4 $ → $ \{4\} $
- Function? Yes — every x maps to one y

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ \{4\} $
- Function? Yes

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9. $ x = y^2 $

- Graph: Parabola opening to the right, vertex at origin
- Domain: $ x \geq 0 $ → $ [0, \infty) $
- Range: All real y-values → $ (-\infty, \infty) $
- Function? No — fails vertical line test (e.g., at $ x = 1 $, two y-values: $ y = 1 $ and $ y = -1 $)

✘ Answer:
- Domain: $ [0, \infty) $
- Range: $ (-\infty, \infty) $
- Function? No

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10. $ y = -|x - 3| + 2 $

- Graph: Inverted V-shape, vertex at $ (3, 2) $
- Domain: All real numbers → $ (-\infty, \infty) $
- Range: Maximum at $ y = 2 $, decreases on both sides → $ (-\infty, 2] $
- Function? Yes

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ (-\infty, 2] $
- Function? Yes

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✔ Final Answers Summary:

| # | Domain | Range | Function? |
|---|-------------------------|--------------------------|-----------|
| 1 | $ (-\infty, \infty) $ | $ (-\infty, \infty) $ | Yes |
| 2 | $ (-\infty, \infty) $ | $ (-\infty, \infty) $ | Yes |
| 3 | $ (-\infty, \infty) $ | $ [-5, \infty) $ | Yes |
| 4 | $ (-\infty, \infty) $ | $ [3, \infty) $ | Yes |
| 5 | $ (-\infty, \infty) $ | $ (-\infty, 3] $ | Yes |
| 6 | $ (-\infty, \infty) $ | $ [-4, \infty) $ | Yes |
| 7 | $ \{-5\} $ | $ (-\infty, \infty) $ | No |
| 8 | $ (-\infty, \infty) $ | $ \{4\} $ | Yes |
| 9 | $ [0, \infty) $ | $ (-\infty, \infty) $ | No |
|10 | $ (-\infty, \infty) $ | $ (-\infty, 2] $ | Yes |

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🔍 Key Concepts Recap:

- Domain: All x-values where the graph exists.
- Range: All y-values that are output.
- Function: Must pass the vertical line test — no vertical line crosses the graph more than once.

Let me know if you'd like this in a printable format!