https://file.aiquickdraw.com/mathbot/file/1726428068170iftlxm11.jpg

Let's break down the two parts of the question, understanding what it means for \( f \) to be a function:

### Part (a):
"Is it possible that \( f(1) = 5 \) and \( f(2) = 5 \)?"

For a function, the only rule that must be followed is that each input can have only **one output**. So, in this case:
- The function assigns 5 to \( f(1) \).
- It also assigns 5 to \( f(2) \).

Since \( f(1) \) and \( f(2) \) are two different inputs (1 and 2), it's perfectly fine for them to both give the same output (5). **This is possible** for a function.

### Conclusion for (a): **Yes**, it is possible that \( f(1) = 5 \) and \( f(2) = 5 \).

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### Part (b):
"Is it possible that \( f(1) = 5 \) and \( f(1) = 6 \)?"

Here, both conditions refer to the same input \( f(1) \), but with two different outputs (5 and 6). For a function, **one input can only have one output**. So, if \( f(1) \) is 5, it cannot also be 6.

### Conclusion for (b): **No**, it is not possible for \( f(1) = 5 \) and \( f(1) = 6 \) to occur at the same time.

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### Summary:
- (a) Yes, it is possible.
- (b) No, it is not possible.

Would you like further clarification or more details?

Here are some related questions to think about:
1. What is the definition of a function in terms of inputs and outputs?
2. Can a function ever have multiple outputs for a single input?
3. What is the difference between a surjective and injective function?
4. If \( f(1) = 5 \) and \( f(2) = 6 \), what does that tell us about the function's nature?
5. Can functions be defined in ways where the output for an input changes?

**Tip:** Always remember that for a function, every input has exactly one output. This is what distinguishes functions from more general relations.

Is it possible that f(1) = 5 and f(2) = 5? Is it possible that f(1) = 5 and f(1) = 6?

Explore the key concepts of functions by answering questions like 'Is it possible that f(1) = 5 and f(2) = 5?' and 'Can f(1) = 5 and f(1) = 6 at the same time?' This exercise helps clarify the foundational rules of functions, ideal for students in Grades 9-12.

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