# function notation

October 23, 2020Uncategorized

with domain X and codomain Y, is bijective, if for every y in Y, there is one and only one element x in X such that y = f(x). ↦

= ⊆ For instance, the following is a piecewise function: As you can see, this function is split into two halves: the half that comes before x = 1, and the half that goes from x = 1 to infinity. )

If you were to write the above information in the form of an expression, it would look something like: Essentially, y is replaced with f(x). and another which is negative and denoted onto its image = E.g., if

⊆ {\displaystyle x^{2}+1} The index notation is also often used for distinguishing some variables called parameters from the "true variables". A binary relation is functional (also called right-unique) if, A binary relation is serial (also called left-total) if. In this case, an element x of the domain is represented by an interval of the x-axis, and the corresponding value of the function, f(x), is represented by a rectangle whose base is the interval corresponding to x and whose height is f(x) (possibly negative, in which case the bar extends below the x-axis).

{\displaystyle (x_{1},\ldots ,x_{n})} , .

A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function.
)

f

, if If the domain is contained in a Euclidean space, or more generally a manifold, a vector-valued function is often called a vector field. 2

is an element of the Cartesian product of copies of )

f R x

u R R

Note: f(x) is the most common way to denote a function, but both the function name and the argument can be changed to any symbol you want. S There are various standard ways for denoting functions. {\displaystyle g(y)=x,}

the preimage The other way is to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity.

=

x (

The most frequently used function notation is f (x) which is read as “f” of “x”. {\displaystyle f\circ g=\operatorname {id} _{Y}.}

{\displaystyle f^{-1}(y)}

and

f

Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. See also Poincaré map. ), URL: https://www.purplemath.com/modules/fcnnot.htm, © 2020 Purplemath. 3

such that

,

∈ f

) 