Introduction to One-to-one Functions

A one-to-one function is a mathematical function in which each element of the domain maps to a distinct and unique element in the range. This property ensures that no two different inputs result in the same output, formally expressed as f (x₁) ≠ f (x₂) whenever x₁ ≠ x₂. The graphical criterion for identifying such functions is the Horizontal Line Test, which indicates that a function is one-to-one if and only if no horizontal line intersects its graph at more than one point.

A quadratic function f (x) = x² is not inherently one-to-one over its entire domain. For instance, both x = 1 and x =-1 yield the same value when squared, violating the one-to-one condition. However, restricting the domain—for example, to non-negative values—can recover the one-to-one property. This makes the function eligible for an inverse.

The inverse of a function, denoted by f ^-1(x), reverses the input-output relationship of the original function. If f (a) = b, then f ^-1(b) = a. In a graph, the inverse function reflects the original function across the line y = x, visually confirming that the function and its inverse have swapped domains and ranges. This symmetry emphasizes the uniqueness and reversibility inherent in one-to-one functions.