One-to-one functions

To make it simple, for the function $f(x)$, all of the possible $x$ values constitute the domain, and all of the values $f(x)$ ($y$ on the x-y plane) constitute the range. To put it in more formal terms, a function $f$ is a mapping of some element $a\in A$, called the domain, to exactly one element $b\in B$, called the range, such that $f:A\to B$. The image below should help explain the modern definition of a function:

$A$ is the domain of the function while $B$ is the range. This transformation from set $A$ to $B$ is an example of one-to-one function.

A function is considered one-to-one if an element $a\in A$ from domain $A$ of function $f$, leads to exactly one element $b\in B$ from range $B$ of the function. By definition, since only one element $b$ is mapped by function $f$ from some element $a$, $f:A\to B$ implies that there exists only one element $b$ from the mapping. Therefore, there exists a one-to-one function because it complies with the definition of a function. This definition is similar to Figure 1.

• One-to-one functions. Written notes created by Professor Esparza, UTSA.