What is what does it mean when a function is one to one?

A function is said to be one-to-one (also called injective) if each element of the range corresponds to exactly one element of the domain.

In simpler terms:

• A function f is one-to-one if for any two distinct elements x1 and x2 in the domain of f, f(x1) is not equal to f(x2). This is the same as saying if f(x1) = f(x2), then x1 = x2.

• Graphically, a function is one-to-one if it passes the horizontal line test. This means that any horizontal line drawn on the graph of the function intersects the graph at most once.

Why is being one-to-one important?

• A one-to-one function has an inverse function. If a function is not one-to-one, it does not have a proper inverse.

• One-to-one functions are important in various areas of mathematics, including set theory, topology, and analysis. They play a crucial role in defining bijections and establishing correspondences between sets.