Math Problem Statement

One-to-one function Definition Example/Symbol

Solution

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:ABf: A \to B is one-to-one if:

f(x1)=f(x2)    x1=x2f(x_1) = f(x_2) \implies x_1 = x_2

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


Example and Symbol

Example:

The function f(x)=2x+3f(x) = 2x + 3 is one-to-one because for any two distinct inputs x1x_1 and x2x_2, f(x1)f(x2)f(x_1) \neq f(x_2).

Symbol:

One-to-one functions are often expressed symbolically using the implication: f(x1)=f(x2)    x1=x2f(x_1) = f(x_2) \implies x_1 = x_2 Alternatively, a function f(x)f(x) is not one-to-one if: x1,x2domain such that f(x1)=f(x2) but x1x2\exists x_1, x_2 \in \text{domain such that } f(x_1) = f(x_2) \text{ but } x_1 \neq x_2


Visual Example:

  • For f(x)=x2f(x) = x^2 (defined on x0x \geq 0), the function is one-to-one.
  • For f(x)=x2f(x) = x^2 (defined for all real xx), it is not one-to-one because f(2)=f(2)=4f(2) = f(-2) = 4.

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.

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Math Problem Analysis

Mathematical Concepts

Functions
Injective Functions
One-to-One Functions

Formulas

f(x1) = f(x2) implies x1 = x2

Theorems

Horizontal Line Test

Suitable Grade Level

Grades 9-12