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### Problem Statement:
You are given a set of relations between \(x\) and \(y\) in the form of mappings and asked to determine if the given relations define \(y\) as a function of \(x\). You must write "F" if it is a function and "R" if it is not a function, and justify your answer.

### Part A: 
The problem asks about the mappings in two diagrams.

#### Diagram 1:
The first diagram has the following mappings:
- \(1 \mapsto 5\)
- \(2 \mapsto 6\)
- \(3 \mapsto 7\)
- \(4 \mapsto 8\)

**Analysis**: 
- In this diagram, each element in the domain (\(x\)) maps to exactly one element in the range (\(y\)).
- This satisfies the definition of a function where every input \(x\) has exactly one output \(y\).

**Conclusion**: The relation is a function. You should write "F" and the justification is that each input has exactly one output.

#### Diagram 2:
The second diagram has the following mappings:
- \(1 \mapsto 5\) and \(1 \mapsto 6\)
- \(2 \mapsto 7\)
- \(3 \mapsto 8\)
- \(4 \mapsto 9\)

**Analysis**: 
- Here, the element \(1\) in the domain maps to two different elements in the range (\(5\) and \(6\)).
- This does not satisfy the definition of a function, which requires each input \(x\) to map to exactly one output \(y\).

**Conclusion**: The relation is **not** a function. You should write "R" and the justification is that the input \(1\) maps to more than one output.

### Additional Questions:
1. What defines a mathematical function?
2. Can a function have the same output for different inputs?
3. What are the consequences if a function maps one input to multiple outputs?
4. How does the vertical line test relate to determining if a relation is a function?
5. What are some examples of real-world situations where functions are applied?

### Tip:
Always remember, a function can map multiple inputs to the same output, but a single input cannot map to multiple outputs.

Understand how to determine if a relation between variables defines a function. Analyze mappings and justify whether each relation is a function or not based on given examples. Suitable for students in Grades 7-9 studying mathematical functions.

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