A function is one-to-one if there exists two values of x that are not the same, and mapped to two values of y that are also not the same. 

The definition you've provided for a **one-to-one function** (also called an **injective** function) needs a slight adjustment for clarity. Here's the correct version:

A function \( f(x) \) is **one-to-one** (or **injective**) if **different** values of \( x \) map to **different** values of \( y \). More formally:

If \( f(x_1) = f(x_2) \), then it must be the case that \( x_1 = x_2 \).

Alternatively, this means that if \( x_1 \neq x_2 \), then \( f(x_1) \neq f(x_2) \). In simpler terms, no two distinct inputs can have the same output for a function to be one-to-one.

Would you like further examples or explanations?

Here are 5 questions that expand on this information:

1. How do you test if a function is one-to-one using its graph?
2. Can a function be one-to-one but not onto (surjective)?
3. What is an example of a real-world application of injective functions?
4. How does the horizontal line test determine if a function is one-to-one?
5. Can an inverse function exist if the original function is not one-to-one?

**Tip:** If you can find two different inputs that give the same output, the function is **not** one-to-one.

A function is one-to-one if there exists two values of x that are not the same, and mapped to two values of y that are also not the same.

Explore the concept of injective or one-to-one functions in mathematics, where different inputs produce different outputs. Learn how to identify these functions using the horizontal line test and the formal definition. Suitable for Grades 10-12 students.

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