Functions And Relations Worksheet Worksheet - Free Printable

Let's go through each part of the Functions and Relations Worksheet step by step.

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Part 1: Decide whether the graph represents $ y $ as a function of $ x $. Explain your reasoning.

We use the Vertical Line Test: If any vertical line intersects the graph more than once, then it is not a function.

#### Graph 1:
- This appears to be a straight line (likely diagonal).
- Any vertical line will intersect it only once.
✔ Yes, this is a function.

> Reasoning: Each $ x $-value has exactly one $ y $-value.

#### Graph 2:
- This looks like a parabola opening upward (U-shaped).
- A vertical line will intersect the graph at most once.
✔ Yes, this is a function.

> Reasoning: Even though it curves, no vertical line crosses it more than once.

#### Graph 3:
- This is a circle.
- A vertical line can pass through two points on the circle (e.g., at $ x = 0 $, there are two $ y $-values).
✘ No, this is not a function.

> Reasoning: For some $ x $-values, there are two corresponding $ y $-values — violates the definition of a function.

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Part 2: Decide whether the relation is a function. If it is, give the domain and range.

#### 4. Input → Output
```
0 → -1
1 → 0
2 → 1
3 → 2
```

- Each input maps to exactly one output.
- No repeated inputs with different outputs.
✔ Yes, this is a function.

- Domain: {0, 1, 2, 3}
- Range: {-1, 0, 1, 2}

#### 5. Input → Output
```
1 → 2
2 → 4
3 → 6
4 → 8
```

- Each input maps to one output.
✔ Yes, this is a function.

- Domain: {1, 2, 3, 4}
- Range: {2, 4, 6, 8}

#### 6. Input → Output
```
0 → 6
4 → -3
```

- Each input has one output.
✔ Yes, this is a function.

- Domain: {0, 4}
- Range: {6, -3}

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Part 3: Evaluate the function when $ x = 3 $, $ x = 0 $, and $ x = -2 $.

We substitute the values into each function.

#### 7. $ f(x) = x $
- $ f(3) = 3 $
- $ f(0) = 0 $
- $ f(-2) = -2 $

#### 8. $ h(x) = x + 7 $
- $ h(3) = 3 + 7 = 10 $
- $ h(0) = 0 + 7 = 7 $
- $ h(-2) = -2 + 7 = 5 $

#### 9. $ g(x) = x + 2 $
- $ g(3) = 3 + 2 = 5 $
- $ g(0) = 0 + 2 = 2 $
- $ g(-2) = -2 + 2 = 0 $

#### 10. $ g(x) = 3x $
- $ g(3) = 3 \times 3 = 9 $
- $ g(0) = 3 \times 0 = 0 $
- $ g(-2) = 3 \times (-2) = -6 $

#### 11. $ g(x) = 4x - 1 $
- $ g(3) = 4(3) - 1 = 12 - 1 = 11 $
- $ g(0) = 4(0) - 1 = 0 - 1 = -1 $
- $ g(-2) = 4(-2) - 1 = -8 - 1 = -9 $

#### 12. $ h(x) = 1.2x $
- $ h(3) = 1.2 \times 3 = 3.6 $
- $ h(0) = 1.2 \times 0 = 0 $
- $ h(-2) = 1.2 \times (-2) = -2.4 $

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✔ Final Answers Summary:

#### Part 1: Graphs
1. Yes – passes vertical line test.
2. Yes – parabola opens up, passes vertical line test.
3. No – circle fails vertical line test.

#### Part 2: Relations
4. Yes; Domain: {0,1,2,3}; Range: {-1,0,1,2}
5. Yes; Domain: {1,2,3,4}; Range: {2,4,6,8}
6. Yes; Domain: {0,4}; Range: {6,-3}

#### Part 3: Function Evaluations

| Function | $ x = 3 $ | $ x = 0 $ | $ x = -2 $ |
|--------|------------|------------|-------------|
| 7. $ f(x) = x $ | 3 | 0 | -2 |
| 8. $ h(x) = x + 7 $ | 10 | 7 | 5 |
| 9. $ g(x) = x + 2 $ | 5 | 2 | 0 |
| 10. $ g(x) = 3x $ | 9 | 0 | -6 |
| 11. $ g(x) = 4x - 1 $ | 11 | -1 | -9 |
| 12. $ h(x) = 1.2x $ | 3.6 | 0 | -2.4 |

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