Five different relationships between z and y are shown. Select all of the relationships that are functions

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### Exploring Functions: Identifying Relationships

Below are five distinct relationships between \( x \) and \( y \). Your task is to determine which of these relationships qualify as functions. Recall that for a relation to be a function, each input value \( x \) should correspond to exactly one output value \( y \).

#### 1. Tabular Representation
The first relationship is represented in a table format:

| \( x \) | \( y \) |
|--------|--------|
| 0      | -2     |
| 1      | -2     |
| 2      | -2     |
| 3      | -2     |

Each input \( x \) (0, 1, 2, 3) maps to exactly one output \( y \) (-2).

#### 2. Graphical Representation (Parabola)
The second relationship is shown as a graph of a quadratic equation:

![Graph](image_url)

This graph displays a parabola opening upwards. Since a vertical line can intersect a parabola in more than one location, it does not meet the criteria for a function.

#### 3. Set of Ordered Pairs
The third relationship is depicted as a collection of ordered pairs:

\[ \{(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), (9, 10)\} \]

Each pair with a unique input \( x \) yields a unique output \( y \).

#### 4. Mapping Diagram
The fourth representation uses a mapping diagram:

![Mapping Diagram](image_url)

The mapping diagram displays inputs \( x \) (1, 2) connected to various outputs \( y \) (-3, -2, -1, 0, 1). Since one input can connect to multiple outputs, this is not a function.

#### 5. Algebraic Expression
The fifth relationship is given as an algebraic equation:

\[ y = 3.5x \]

This equation indicates a linear relationship where each \( x \) value corresponds to exactly one \( y \) value.

### Analysis and Identification

From these representations, identify which among them qualify as functions based on the definition of a function. Select all that apply.
Transcribed Image Text:### Exploring Functions: Identifying Relationships Below are five distinct relationships between \( x \) and \( y \). Your task is to determine which of these relationships qualify as functions. Recall that for a relation to be a function, each input value \( x \) should correspond to exactly one output value \( y \). #### 1. Tabular Representation The first relationship is represented in a table format: | \( x \) | \( y \) | |--------|--------| | 0 | -2 | | 1 | -2 | | 2 | -2 | | 3 | -2 | Each input \( x \) (0, 1, 2, 3) maps to exactly one output \( y \) (-2). #### 2. Graphical Representation (Parabola) The second relationship is shown as a graph of a quadratic equation: ![Graph](image_url) This graph displays a parabola opening upwards. Since a vertical line can intersect a parabola in more than one location, it does not meet the criteria for a function. #### 3. Set of Ordered Pairs The third relationship is depicted as a collection of ordered pairs: \[ \{(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), (9, 10)\} \] Each pair with a unique input \( x \) yields a unique output \( y \). #### 4. Mapping Diagram The fourth representation uses a mapping diagram: ![Mapping Diagram](image_url) The mapping diagram displays inputs \( x \) (1, 2) connected to various outputs \( y \) (-3, -2, -1, 0, 1). Since one input can connect to multiple outputs, this is not a function. #### 5. Algebraic Expression The fifth relationship is given as an algebraic equation: \[ y = 3.5x \] This equation indicates a linear relationship where each \( x \) value corresponds to exactly one \( y \) value. ### Analysis and Identification From these representations, identify which among them qualify as functions based on the definition of a function. Select all that apply.
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