Solved Function Notation and Evaluating Functions Pracfice | Chegg.com - Free Printable

Problem Overview:

The task involves analyzing graphs and relations to determine whether they represent functions. A function is a special type of relation where each input (x-value) corresponds to exactly one output (y-value). If a relation does not satisfy this condition, it is simply a relation.

We will analyze each graph and relation step by step:

---

Part 1: Graphs (Questions 1–6)

#### Question 1:
- Graph Description: The graph shows multiple points scattered across the plane.
- Vertical Line Test: Draw vertical lines across the graph. If any vertical line intersects the graph at more than one point, the graph is not a function.
- Analysis: Vertical lines can intersect the graph at multiple points.
- Conclusion: This is not a function.

#### Question 2:
- Graph Description: The graph shows a parabola opening upwards.
- Vertical Line Test: Any vertical line will intersect the parabola at most once.
- Analysis: The graph passes the vertical line test.
- Conclusion: This is a function.
- Domain: All real numbers, \( (-\infty, \infty) \).
- Range: Since the parabola opens upwards and its vertex is at the origin, the range is \( [0, \infty) \).

#### Question 3:
- Graph Description: The graph shows an "X" shape, which is a hyperbola.
- Vertical Line Test: Vertical lines can intersect the graph at two points.
- Analysis: The graph fails the vertical line test.
- Conclusion: This is not a function.

#### Question 4:
- Graph Description: The graph shows a V-shaped figure (absolute value function).
- Vertical Line Test: Any vertical line will intersect the graph at most once.
- Analysis: The graph passes the vertical line test.
- Conclusion: This is a function.
- Domain: All real numbers, \( (-\infty, \infty) \).
- Range: The lowest point of the V-shape is at \( y = 0 \), so the range is \( [0, \infty) \).

#### Question 5:
- Graph Description: The graph shows a smooth, wavy curve.
- Vertical Line Test: Any vertical line will intersect the graph at most once.
- Analysis: The graph passes the vertical line test.
- Conclusion: This is a function.
- Domain: All real numbers, \( (-\infty, \infty) \).
- Range: The curve oscillates between certain minimum and maximum values. From the graph, it appears the range is approximately \( [-1, 1] \).

#### Question 6:
- Graph Description: The graph shows a horizontal line.
- Vertical Line Test: Any vertical line will intersect the graph at most once.
- Analysis: The graph passes the vertical line test.
- Conclusion: This is a function.
- Domain: All real numbers, \( (-\infty, \infty) \).
- Range: The horizontal line is at \( y = -2 \), so the range is \( \{-2\} \).

---

Part 2: Relations (Questions 7–9)

#### Question 7:
- Relation Description:
- Input: \( 1 \rightarrow 7 \)
- Input: \( 2 \rightarrow -8 \)
- Input: \( 2 \rightarrow -8 \)
- Input: \( 1 \rightarrow -7 \)
- Analysis: The input \( 1 \) maps to both \( 7 \) and \( -7 \), and the input \( 2 \) maps to both \( 8 \) and \( -8 \).
- Conclusion: This is not a function because inputs are not uniquely mapped to outputs.

#### Question 8:
- Relation Description:
- Input: \( 3 \rightarrow 2 \)
- Input: \( 5 \rightarrow 4 \)
- Input: \( 7 \rightarrow 6 \)
- Analysis: Each input maps to exactly one output.
- Conclusion: This is a function.
- Domain: \( \{3, 5, 7\} \)
- Range: \( \{2, 4, 6\} \)

#### Question 9:
- Relation Description:
- Input: \( 0 \rightarrow -6 \)
- Input: \( 2 \rightarrow -4 \)
- Input: \( 4 \rightarrow -2 \)
- Input: \( 6 \rightarrow 0 \)
- Analysis: Each input maps to exactly one output.
- Conclusion: This is a function.
- Domain: \( \{0, 2, 4, 6\} \)
- Range: \( \{-6, -4, -2, 0\} \)

---

Final Answers:

1. Not a function.
2. Function; Domain: \( (-\infty, \infty) \), Range: \( [0, \infty) \).
3. Not a function.
4. Function; Domain: \( (-\infty, \infty) \), Range: \( [0, \infty) \).
5. Function; Domain: \( (-\infty, \infty) \), Range: \( [-1, 1] \).
6. Function; Domain: \( (-\infty, \infty) \), Range: \( \{-2\} \).
7. Not a function.
8. Function; Domain: \( \{3, 5, 7\} \), Range: \( \{2, 4, 6\} \).
9. Function; Domain: \( \{0, 2, 4, 6\} \), Range: \( \{-6, -4, -2, 0\} \).

Boxed Final Answer:
\[
\boxed{
\begin{array}{ll}
1. & \text{Not a function.} \\
2. & \text{Function; Domain: } (-\infty, \infty), \text{ Range: } [0, \infty). \\
3. & \text{Not a function.} \\
4. & \text{Function; Domain: } (-\infty, \infty), \text{ Range: } [0, \infty). \\
5. & \text{Function; Domain: } (-\infty, \infty), \text{ Range: } [-1, 1]. \\
6. & \text{Function; Domain: } (-\infty, \infty), \text{ Range: } \{-2\}. \\
7. & \text{Not a function.} \\
8. & \text{Function; Domain: } \{3, 5, 7\}, \text{ Range: } \{2, 4, 6\}. \\
9. & \text{Function; Domain: } \{0, 2, 4, 6\}, \text{ Range: } \{-6, -4, -2, 0\}.
\end{array}
}
\]