Functions

A relation between two sets $X$ and $Y$ is a set of ordered pairs $(x , y),$ where $x$ is an element of $X$ and $y$ is an element of $Y.$

Let $X$ and $Y$ be sets of real numbers. A real-valued function $f$ of a real variable $x$ from $X$ to $Y$ is a relation that assigns to each number $x$ in $X$ exactly one number $y$ in $Y.$ The domain of $f$ is the set $X.$ The codomain of $f$ is the set $Y.$ The number $y$ is the image of $x$ under $f$ and is denoted $f(x).$ The range of $f$ is a subset of $Y$ and consists of all images of numbers in $X.$ The variable $x$ is the independant variable, and the variable $y$ is the dependant variable.

A function is one-to-one if to each $y$ -value in the range there corresponds exactly one $x$ -value in the domain. $f$ is one-to-one if $a\ne b \implies f(a)\ne f(b).$