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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.