https://file.aiquickdraw.com/mathbot/file/1729115676223ux1hx2qo.jpg

To determine if each relation is a function, we need to check whether each element in the domain maps to *only one* element in the range. If any element in the domain maps to multiple elements in the range, then it is not a function.

Let's analyze each relation:

### Relation 1:
- Domain: \( e, t, k, m \)
- Range: \( n, t, t, n \)

Each element in the domain maps to a single, unique element in the range:
- \( e \to n \)
- \( t \to t \)
- \( k \to t \)
- \( m \to n \)

Since each domain element has only one corresponding range element, **Relation 1 is a function**.

### Relation 2:
- Domain: \( -9, -7, 2, 6, -3 \)
- Range: \( \text{lake, sky, rock, star, chair} \)

Each element in the domain maps to one unique element in the range:
- \( -9 \to \text{lake} \)
- \( -7 \to \text{sky} \)
- \( 2 \to \text{rock} \)
- \( 6 \to \text{star} \)
- \( -3 \to \text{chair} \)

Since each domain element has a unique range element, **Relation 2 is a function**.

### Relation 3:
- Set of pairs: \( \{ (8, 8), (8, 1), (8, 5), (8, 0) \} \)

Here, the domain element \( 8 \) maps to multiple range values \( 8, 1, 5, \) and \( 0 \), which violates the definition of a function. Each input (domain element) must map to only one output.
Therefore, **Relation 3 is not a function**.

### Relation 4:
- Set of pairs: \( \{ (g, 4), (c, 5), (k, -8), (t, 1) \} \)

Each element in the domain has a unique range value:
- \( g \to 4 \)
- \( c \to 5 \)
- \( k \to -8 \)
- \( t \to 1 \)

Since each domain element has only one corresponding range element, **Relation 4 is a function**.

### Summary:
1. Relation 1: **Function**
2. Relation 2: **Function**
3. Relation 3: **Not a function**
4. Relation 4: **Function**

Would you like further details on any of these relations?

#### Additional Questions:
1. What is the main characteristic that defines a function in terms of domain and range?
2. How does the vertical line test help in identifying functions graphically?
3. Can a relation with repeated elements in the range still be a function?
4. What are examples of real-world situations modeled by functions?
5. How would the analysis change if the relations were represented by equations?

#### Tip:
Remember, to be a function, each input in the domain must correspond to exactly one output in the range.

For each relation, decide whether or not it is a function. (Relations are provided in the image, showing tables and sets of ordered pairs.)

Analyze relations to determine if they are functions based on their domain and range. Understand key math concepts, including the definition of a function, with step-by-step explanations. Suitable for students in Grades 8-10.

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