In mathematics, a function $f(x)$ is called one-to-one (or injective) if it satisfies the property that different inputs produce different outputs. This means that if the function produces the same output for two inputs, then those two inputs must be the same.

To understand this more clearly, consider the following logical steps:

1. Assume $f(x_1) = f(x_2)$: This means that the outputs of the function for the inputs $x_1$ and $x_2$ are equal.

2. By definition of one-to-one functions: A function is one-to-one if it guarantees that for any two inputs, if the outputs are the same, then the inputs must also be the same. In symbols, if $f(x_1) = f(x_2)$, it implies that $x_1 = x_2$.

Thus, for a function to be one-to-one, $f(x_1) = f(x_2)$ implies $x_1 = x_2$.

Explanation in steps:

• $f(x_1) = f(x_2)$: We start with the assumption that the function gives the same result for both $x_1$ and $x_2$.
• Since the function is one-to-one, it is defined in such a way that equal outputs only occur if the inputs are the same.
• Therefore, the assumption that $f(x_1) = f(x_2)$ leads to the conclusion that $x_1 = x_2$.

This property prevents different values of $x$ from mapping to the same value of $f(x)$, ensuring the function is injective.

Would you like more examples or further clarification on injective functions?

Here are some related questions for deeper understanding:

1. How can you prove that a function is one-to-one?
2. What are some real-world examples of one-to-one functions?
3. How do you check if a function is one-to-one using its derivative?
4. What’s the difference between one-to-one and onto functions?
5. Can a function be both one-to-one and many-to-one at the same time?

Tip: You can verify if a function is one-to-one by using the horizontal line test: if every horizontal line intersects the graph of the function at most once, the function is injective.