Independent Practice Domain Range End Behavior Answer Key ... - Free Printable

Problem Analysis:

The image contains two main tasks:
1. Determining if the given relations are functions (Section 1).
2. Writing inequalities based on number lines (Section 2).

Let's solve each section step by step.

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Section 1: Are the following relations functions? Why or why not?

#### Key Concept:
A relation is a function if and only if each input (x-value) is associated with exactly one output (y-value). In other words, no x-value can be paired with more than one y-value.

#### Analysis of Each Relation:

##### (a)
- The relation is represented as a set of ordered pairs: `{(2, 3), (4, 5), (6, 7), (8, 9)}`.
- Each x-value (2, 4, 6, 8) is paired with exactly one y-value.
- Conclusion: This is a function.

##### (b)
- The relation is represented as a set of ordered pairs: `{(2, 3), (4, 5), (6, 7), (8, 9)}`.
- Each x-value (2, 4, 6, 8) is paired with exactly one y-value.
- Conclusion: This is a function.

##### (c)
- The relation is represented as a mapping diagram where each x-value maps to exactly one y-value.
- Conclusion: This is a function.

##### (d)
- The relation is represented as a mapping diagram where one x-value (e.g., 4) maps to multiple y-values (e.g., 6 and 8).
- Conclusion: This is not a function.

##### (e)
- The relation is represented as a graph.
- The graph passes the vertical line test: any vertical line drawn will intersect the graph at most once.
- Conclusion: This is a function.

##### (f)
- The relation is represented as a graph.
- The graph does not pass the vertical line test: a vertical line can intersect the graph at more than one point.
- Conclusion: This is not a function.

##### (g)
- The relation is represented as a graph.
- The graph passes the vertical line test: any vertical line drawn will intersect the graph at most once.
- Conclusion: This is a function.

##### (h)
- The relation is represented as a graph.
- The graph passes the vertical line test: any vertical line drawn will intersect the graph at most once.
- Conclusion: This is a function.

#### Final Answers for Section 1:
- (a): Function
- (b): Function
- (c): Function
- (d): Not a function
- (e): Function
- (f): Not a function
- (g): Function
- (h): Function

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Section 2: Write the interval shown on the number lines as an inequality, using set notation and using interval notation.

#### Key Concepts:
1. Inequality: A mathematical statement that uses symbols like `<`, `>`, `≤`, or `≥`.
2. Set Notation: Represents the solution set using braces `{ }` and describes the elements.
3. Interval Notation: Uses parentheses `( )` or brackets `[ ]` to represent ranges of values.

#### Analysis of Each Number Line:

##### (a)
- The number line shows a closed circle at `-4` and an arrow pointing to the left.
- This indicates all values less than or equal to `-4`.
- Inequality: \( x \leq -4 \)
- Set Notation: \( \{ x \mid x \leq -4 \} \)
- Interval Notation: \( (-\infty, -4] \)

##### (b)
- The number line shows an open circle at `0` and an arrow pointing to the right.
- This indicates all values greater than `0`.
- Inequality: \( x > 0 \)
- Set Notation: \( \{ x \mid x > 0 \} \)
- Interval Notation: \( (0, \infty) \)

##### (c)
- The number line shows a closed circle at `-2` and an arrow pointing to the right.
- This indicates all values greater than or equal to `-2`.
- Inequality: \( x \geq -2 \)
- Set Notation: \( \{ x \mid x \geq -2 \} \)
- Interval Notation: \( [-2, \infty) \)

##### (d)
- The number line shows a closed circle at `3` and an arrow pointing to the left.
- This indicates all values less than or equal to `3`.
- Inequality: \( x \leq 3 \)
- Set Notation: \( \{ x \mid x \leq 3 \} \)
- Interval Notation: \( (-\infty, 3] \)

#### Final Answers for Section 2:
- (a):
- Inequality: \( x \leq -4 \)
- Set Notation: \( \{ x \mid x \leq -4 \} \)
- Interval Notation: \( (-\infty, -4] \)

- (b):
- Inequality: \( x > 0 \)
- Set Notation: \( \{ x \mid x > 0 \} \)
- Interval Notation: \( (0, \infty) \)

- (c):
- Inequality: \( x \geq -2 \)
- Set Notation: \( \{ x \mid x \geq -2 \} \)
- Interval Notation: \( [-2, \infty) \)

- (d):
- Inequality: \( x \leq 3 \)
- Set Notation: \( \{ x \mid x \leq 3 \} \)
- Interval Notation: \( (-\infty, 3] \)

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Final Answer:

\[
\boxed{
\text{Section 1: (a) Function, (b) Function, (c) Function, (d) Not a function, (e) Function, (f) Not a function, (g) Function, (h) Function}
}
\]
\[
\boxed{
\text{Section 2: (a) } x \leq -4, \{ x \mid x \leq -4 \}, (-\infty, -4]; \text{ (b) } x > 0, \{ x \mid x > 0 \}, (0, \infty); \text{ (c) } x \geq -2, \{ x \mid x \geq -2 \}, [-2, \infty); \text{ (d) } x \leq 3, \{ x \mid x \leq 3 \}, (-\infty, 3]
}
\]