Definition: One-to-One Function

A one-to-one function (or injective function) is a type of function where each element of the domain maps to a unique element of the codomain. In other words, no two different inputs in the domain produce the same output in the codomain. Mathematically, a function $f: A \to B$ is one-to-one if:

$f(x_1) = f(x_2) \implies x_1 = x_2$

This ensures that the function does not "overlap" values in its range.

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Example and Symbol

Example:

The function $f(x) = 2x + 3$ is one-to-one because for any two distinct inputs $x_1$ and $x_2$, $f(x_1) \neq f(x_2)$.

Symbol:

One-to-one functions are often expressed symbolically using the implication: $f(x_1) = f(x_2) \implies x_1 = x_2$ Alternatively, a function $f(x)$ is not one-to-one if: $\exists x_1, x_2 \in \text{domain such that } f(x_1) = f(x_2) \text{ but } x_1 \neq x_2$

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Visual Example:

• For $f(x) = x^2$ (defined on $x \geq 0$), the function is one-to-one.
• For $f(x) = x^2$ (defined for all real $x$), it is not one-to-one because $f(2) = f(-2) = 4$.

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Would you like further clarification or a detailed proof of the example?

Relative Questions

1. How do we determine if a function is one-to-one using derivatives?
2. Can a one-to-one function also be onto? What is the relation?
3. How can we test if a function is one-to-one using the horizontal line test?
4. Are all linear functions one-to-one? Why or why not?
5. How does restricting the domain of a function help make it one-to-one?

Tip: Use the horizontal line test: If a horizontal line intersects the graph of a function more than once, the function is not one-to-one.