function

in mathematics, a function is a relation that is either a one-to-one relation or a many-to-one relation

glossary

• relation
• set

introduction

a function is a relation with one input set $X$ and one output set $Y$
set $X$ is called the domain and set $Y$ is called the codomain or range
in other words, a function from a set $X$ to a set $Y$ assigns to each element of $X$

let two sets $X$ and $Y$ be given. let $x\in X$ and $y\in Y$
a function $f$ is given on $X$ and the variable $y$ is a function of the variable $x$, therefore can be shown as $f(x)$

a function $f(x)$ is defined with the use of equations and cannot be defined with

for example, if a function $f$ inputs $x$ and outputs $x^2$, the formal representation is $f:\mathbb{R}\to [0,\infty ),x\to f(x),f(x)=x^2$
$f$ is the name of the function
$\mathbb{R}$ is the domain of the function
$[0,\infty )$ is the codomain of the function
$x$ is the input of the function
$f(x)$ is the output of the function
and finally, the input and output are related with the equation $f(x)=x^2$

for short, we often just say $f(x)=x^2$ and everything else in the function syntax are implied

composite functions

multiple functions can be combined in function composition

multivariate functions

a multivariate function, also called a multivariable function,
is a relation that inputs or outputs more than one variable
a function that has more than one input or output technically does not count as a function

inverse functions

a function $f$ has an inverse function $f^{-1}$ that can map the codomain onto the domain

derivatives

the derivative of a function $f(x)$ is a function $g(x)$ such that for any constant $c$ , $g(c)$ is equal to the slope of the tangent line at $f(c)$

exponential function

a exponential function

logarithmic function

a logarithmic function

power function

a power function

polynomials

a polynomial