# Function

DEFINITION1: Let $X$ and $Y$ be two sets and $f\subset{X\times{Y}}$ be a relation. If the following two conditions are provided, then the relation $f$ is called a “function” with domain $X$ and codomain $Y$ and denoted by $f:X\to{Y}$ or $X\stackrel{f}{\rightarrow}{Y}$.

1. $\forall{x}\in{X}, \exists{y}\in{Y}: (x,y)\in{f}$,

2. $(x,y),(x,y')\in{f}\Rightarrow{y=y'}$.

Henceforth, when we write $f:X\to{Y}$, we will consider that “$f$ is a function from $X$ to $Y$”.

As is seen from the definition, a function is a relation mapping each element in the domain to a unique element in the codomain. So, the notation $y=f(x)$ is generally used instead of the notations $xfy$ and $(x,y)\in{f}$ and read “$x$ maps to $y$” or “$x$ maps to $f(x)$”. The notation $f(x)$ is read “$f$ of $x$”. Each element of the domain is called an “argument” and for each $x$ in the domain, the corresponding unique element $y$ in the codomain is called “the function value at $x$”, “output $f$ for an element $x$” or “the image $x$” under the function $f$. The set defined as $\{f(x)\:|\:x\in{X}\}\subset{Y}$ is called the “image” or the “range” of $f$. Sometimes a function is called a “map” or a “mapping”.