Domain and Range of Graphs Practice Worksheet ANSWERS | PDF ... - Free Printable

Let's go through each of the 9 graphs one by one, determine their domain, range, and whether they represent a function (using the vertical line test). I'll explain each step clearly.

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1) Graph 1

- Graph: Discrete points plotted at: (-3, -4), (-2, 0), (-1, 5), (0, 3), (1, 4), (2, 0), (3, -2)
- Domain: The set of all x-values:
→ $ x = \{-3, -2, -1, 0, 1, 2, 3\} $
But in the problem, it says: (-3, 5, -2, 4) — this seems to be a typo or misordering.
Looking at the graph, correct domain is: $ x \in \{-3, -2, -1, 0, 1, 2, 3\} $
However, the answer given says: Domain: {x = -3, 5, -2, 4} — that’s incorrect because 5 and 4 are not x-values.

✔ Correct Domain: $ \{-3, -2, -1, 0, 1, 2, 3\} $

- Range: y-values: $ \{-4, -2, 0, 3, 4, 5\} $
Given range: {-4, -2, 0, 3, 5} — missing 4? Wait, point (1,4) exists → so 4 should be included.

✔ Correct Range: $ \{-4, -2, 0, 3, 4, 5\} $

- Function?
Each x has only one y → Yes, it's a function.
But the answer says No → ✘ Incorrect

✔ Answer should be: YES

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2) Graph 2

- Shape: A diamond (rhombus) centered at origin with vertices at (±3,0), (0,±3)
- Domain: x from -3 to 3 → $ -3 \leq x \leq 3 $
- Range: y from -3 to 3 → $ -3 \leq y \leq 3 $
But the problem says: Range: $ -4 \leq x \leq 3 $ — this is wrong! It's mixing up variables.

✔ Correct Range: $ -3 \leq y \leq 3 $

- Function?
Use vertical line test: a vertical line at x=0 hits two points (top and bottom) → fails → Not a function

✔ Answer: No → Correct

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3) Graph 3

- Graph: A piecewise graph starting at open circle at (4, 2), going down to (5, 1), then up to (6, 2), then a curve rising to (7, 3)
- Domain: x > 4 → $ x > 4 $ (open at 4)
- Range: y ≥ 1 → $ y \geq 1 $ (minimum at y=1)
- Function?
Every x has only one y → passes vertical line test → Yes

✔ All answers are correct as given:
- Domain: $ x > 4 $
- Range: $ y \geq 1 $
- Function: Yes

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4) Graph 4

- Graph: Circle centered at (0, 2), radius 2 → equation: $ x^2 + (y - 2)^2 = 4 $
- Domain: x from -2 to 2 → $ -2 \leq x \leq 2 $
- Range: y from 0 to 4 → $ 0 \leq y \leq 4 $
- Function?
Vertical line test: at x=0, two y-values (top and bottom of circle) → Not a function

Given answer: Function? Yes → ✘ Incorrect

✔ Should be: No

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5) Graph 5

- Graph: Straight diagonal line going from bottom-left to top-right, passing through origin
- Domain: All real numbers → $ \mathbb{R} $
- Range: All real numbers → $ \mathbb{R} $
- Function?
Line passes vertical line test → Yes

✔ All answers correct:
- Domain: $ \mathbb{R} $
- Range: $ \mathbb{R} $
- Function: Yes

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6) Graph 6

- Graph: V-shaped graph opening upward, vertex at (0, -5), symmetric about y-axis
- Domain: All real numbers → $ \mathbb{R} $
- Range: y ≥ -5 → $ y \geq -5 $
- Function?
Vertical line test: each x has one y → Yes

✔ All correct:
- Domain: $ \mathbb{R} $
- Range: $ y \geq -5 $
- Function: Yes

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7) Graph 7

- Graph: Parabola-like shape opening right, starting at origin (0,0), two branches going up and down
- Domain: x ≥ 0 → correct
- Range: All real numbers → $ \mathbb{R} $
- Function?
At any x > 0, there are two y-values (one above, one below) → fails vertical line test → Not a function

Given answer: Function? No → ✔ Correct

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8) Graph 8

- Graph: Two horizontal lines:
- One at y = 1, extending left and right
- One at y = 3, extending left and right
- So it's like two parallel horizontal lines
- Domain: All real numbers → $ \mathbb{R} $
- Range: y = 1 or y = 3 → $ \{1, 3\} $
But written as: (y = 1,3) — probably means $ y = 1 $ or $ y = 3 $
- Function?
For any x, there are two y-values (unless restricted), so fails vertical line test → Not a function

Given answer: Function? No → ✔ Correct

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9) Graph 9

- Graph: Piecewise graph with:
- From x = -4 to x = -3: line segment from (-4, 0) to (-3, 0)
- Open circle at (-3, 0), then line from (-3, -2) to (0, 0)
- Then from (0,0) to (2, 1), but with open circle at x=2
- Domain restriction: $ x < 2 $ and $ x \ne -3 $
- Domain: $ x < 2 $ and $ x \ne -3 $ → correct
- Range: Lowest point is y = -2 (at x = -3), and goes up to just under y = 1 → $ y \geq -2 $
- Function?
Each x has only one y → passes vertical line test → Yes

✔ All answers correct:
- Domain: $ x < 2 $, $ x \ne -3 $
- Range: $ y \geq -2 $
- Function: Yes

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## ✔ Final Summary Table:

| # | Domain | Range | Function? | Correct? |
|---|--------|-------|-----------|---------|
| 1 | {-3,-2,-1,0,1,2,3} | {-4,-2,0,3,4,5} | Yes | ✘ Given: No → Incorrect |
| 2 | $-3 \leq x \leq 3$ | $-3 \leq y \leq 3$ | No | ✔ Correct |
| 3 | $x > 4$ | $y \geq 1$ | Yes | ✔ Correct |
| 4 | $-2 \leq x \leq 2$ | $0 \leq y \leq 4$ | No | ✘ Given: Yes → Incorrect |
| 5 | $\mathbb{R}$ | $\mathbb{R}$ | Yes | ✔ Correct |
| 6 | $\mathbb{R}$ | $y \geq -5$ | Yes | ✔ Correct |
| 7 | $x \geq 0$ | $\mathbb{R}$ | No | ✔ Correct |
| 8 | $\mathbb{R}$ | $\{1,3\}$ | No | ✔ Correct |
| 9 | $x < 2$, $x \ne -3$ | $y \geq -2$ | Yes | ✔ Correct |

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## 🔍 Key Concepts:
- Domain: All possible input values (x-values)
- Range: All possible output values (y-values)
- Function: Passes vertical line test — no vertical line intersects graph more than once

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🛠️ Corrections Needed:

- Problem 1: Function? → Should be Yes
- Problem 4: Function? → Should be No

All others are correctly answered.

✔ Final Answers:
1. Function: Yes
2. Correct
3. Correct
4. Function: No
5. Correct
6. Correct
7. Correct
8. Correct
9. Correct

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