Functions and Relations Worksheet featuring graph analysis and function evaluation exercises.

Problem Analysis and Solution

The worksheet focuses on understanding functions, their representations, and evaluating them. Let's solve each part step by step.

---

#### Part 1: Decide whether the graph represents \( y \) as a function of \( x \). Explain your reasoning.

A relation is a function if every \( x \)-value (input) corresponds to exactly one \( y \)-value (output). This can be tested using the vertical line test: if any vertical line intersects the graph at more than one point, the graph does not represent a function.

##### Problem 1:
- Graph Description: The graph is a straight line.
- Vertical Line Test: Any vertical line will intersect the graph at exactly one point.
- Conclusion: The graph represents \( y \) as a function of \( x \).

##### Problem 2:
- Graph Description: The graph is a parabola opening downwards.
- Vertical Line Test: Any vertical line will intersect the graph at exactly one point.
- Conclusion: The graph represents \( y \) as a function of \( x \).

##### Problem 3:
- Graph Description: The graph is a circle.
- Vertical Line Test: A vertical line passing through the center of the circle will intersect the graph at two points.
- Conclusion: The graph does not represent \( y \) as a function of \( x \).

---

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

A relation is a function if each input (\( x \)) is paired with exactly one output (\( y \)).

##### Problem 4:
- Relation:
- Input: \( 0 \), Output: \( 6 \)
- Input: \( 0 \), Output: \( -6 \)
- Input: \( 1 \), Output: \( 5 \)
- Input: \( 1 \), Output: \( -5 \)
- Analysis: The input \( 0 \) is paired with both \( 6 \) and \( -6 \), and the input \( 1 \) is paired with both \( 5 \) and \( -5 \).
- Conclusion: This is not a function.

##### Problem 5:
- Relation:
- Input: \( 1 \), Output: \( 4 \)
- Input: \( 2 \), Output: \( 4 \)
- Input: \( 3 \), Output: \( 4 \)
- Input: \( 4 \), Output: \( 4 \)
- Analysis: Each input is paired with exactly one output (\( 4 \)).
- Conclusion: This is a function.
- Domain: \( \{1, 2, 3, 4\} \)
- Range: \( \{4\} \)

##### Problem 6:
- Relation:
- Input: \( 0 \), Output: \( 6 \)
- Input: \( 2 \), Output: \( 6 \)
- Input: \( 4 \), Output: \( -3 \)
- Analysis: Each input is paired with exactly one output.
- Conclusion: This is a function.
- Domain: \( \{0, 2, 4\} \)
- Range: \( \{6, -3\} \)

---

#### Part 3: Evaluate the function when \( x = 3 \), \( x = 0 \), and \( x = -2 \).

We are given several functions and need to evaluate them at the specified \( x \)-values.

##### Problem 7: \( f(x) = x \)
- Evaluation:
- \( f(3) = 3 \)
- \( f(0) = 0 \)
- \( f(-2) = -2 \)

##### Problem 8: \( h(x) = x + 7 \)
- Evaluation:
- \( h(3) = 3 + 7 = 10 \)
- \( h(0) = 0 + 7 = 7 \)
- \( h(-2) = -2 + 7 = 5 \)

##### Problem 9: \( g(x) = x - 2 \)
- Evaluation:
- \( g(3) = 3 - 2 = 1 \)
- \( g(0) = 0 - 2 = -2 \)
- \( g(-2) = -2 - 2 = -4 \)

##### Problem 10: \( g(x) = 3x \)
- Evaluation:
- \( g(3) = 3 \cdot 3 = 9 \)
- \( g(0) = 3 \cdot 0 = 0 \)
- \( g(-2) = 3 \cdot (-2) = -6 \)

##### Problem 11: \( g(x) = 4x - 1 \)
- Evaluation:
- \( g(3) = 4 \cdot 3 - 1 = 12 - 1 = 11 \)
- \( g(0) = 4 \cdot 0 - 1 = 0 - 1 = -1 \)
- \( g(-2) = 4 \cdot (-2) - 1 = -8 - 1 = -9 \)

##### Problem 12: \( h(x) = 1.2x \)
- Evaluation:
- \( h(3) = 1.2 \cdot 3 = 3.6 \)
- \( h(0) = 1.2 \cdot 0 = 0 \)
- \( h(-2) = 1.2 \cdot (-2) = -2.4 \)

---

Final Answers:

1. Graph 1: Function
2. Graph 2: Function
3. Graph 3: Not a function
4. Relation 4: Not a function
5. Relation 5: Function, Domain: \( \{1, 2, 3, 4\} \), Range: \( \{4\} \)
6. Relation 6: Function, Domain: \( \{0, 2, 4\} \), Range: \( \{6, -3\} \)
7. \( f(x) = x \): \( f(3) = 3 \), \( f(0) = 0 \), \( f(-2) = -2 \)
8. \( h(x) = x + 7 \): \( h(3) = 10 \), \( h(0) = 7 \), \( h(-2) = 5 \)
9. \( g(x) = x - 2 \): \( g(3) = 1 \), \( g(0) = -2 \), \( g(-2) = -4 \)
10. \( g(x) = 3x \): \( g(3) = 9 \), \( g(0) = 0 \), \( g(-2) = -6 \)
11. \( g(x) = 4x - 1 \): \( g(3) = 11 \), \( g(0) = -1 \), \( g(-2) = -9 \)
12. \( h(x) = 1.2x \): \( h(3) = 3.6 \), \( h(0) = 0 \), \( h(-2) = -2.4 \)

\[
\boxed{
\text{See detailed solutions above.}
}
\]