# Some definitions regarding functions

Up a level : Algebra and Arithmetic
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Next page : Properties of Functions Functions of more than one  variable

If we have more than one variable we can bundle these together in one variable containing them all. Let us say we have n variables, x1, x2, …, xn, then we create an n-tuple x=(x1, x2, …, xn,) witch we can see as one variable. If the function could be explicitly defined we usually write this as,

y=f(x1, x2, …, xn,)

or as

y=f (x)

The domain is now the product set of all domains of the variables.

Ordered n-tuples of functions

We can regard  two functions,    f : AB, C → D,  as one,

h : A×CB×D, where h=(f, g)

This type of functions is used in for example vector algebra and vector analysis.
The set product  A×C will be the set of all ordered pairs of elements a from A and b from b, (a, b).

Composite  functions

If we have a function : AB, and a function g : BC, then the function

g (f ) : AC

is the function we get by first applying the function f then the function g. If the functions are explicit we can write

y=f (g (x))

One can often see this written as

y=(f g)(x)

or as

y=(f (g))(x)

Say we have f (x)=2x+3 and g (x)=x2+x, then we have that

f (g (x))=f (x2+x)=2(x2+x)+3=2x2+2x+3

but

g (f (x))=g (2x+3)=(2x+3)2+(2x+3)
=4x2+12x+9+2x+3
=4x2+14x+12

Arithmetic on functions

What does (f+g)(x) mean? It is usually defined as f (x)+g(x), and in general we have

(f*g)(x)=f (x)*g(x)

where ‘*’ stand for any binary relation.

Operators

A function without variables is called an operator. We could thus talk about a square operator or a square root operator.

Arithmetic as functions

The arithmetic operations are functions. Each of them can be seen as a function of one or two variables or, in the later case, one two-tuple variable. We could for example write,  y=a+b, as,

p(a,b)=a+b

We have such functions in for example Excel where the SUM operator is doing the above + that it can add more than two numbers. Up a level : Algebra and Arithmetic
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Next page : Properties of Functions Last modified: Jan 5, 2019 @ 14:04