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**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, *x*_{1}, *x*_{2}, …, *x _{n}*, then we create an n-tuple

*=(*

**x***x*

_{1},

*x*

_{2}, …,

*x*,) witch we can see as one variable. If the function could be explicitly defined we usually write this as,

_{n}*y=f(x _{1}, x_{2}, …, x_{n},) *

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 *: *A* →*B, *C → D, as one,

*h *: *A×C*→*B×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 *f *: *A* →*B*, and a function *g *: *B*→*C*, then the function

*g *(*f *) : *A*→*C*

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*)=2*x*+3 and *g* (*x*)=*x*^{2}+*x*, then we have that

*f *(*g *(*x*))=*f *(*x ^{2}+x*)=2(

*x*

^{2}+

*x*)+3=2

*x*

^{2}+2

*x*+3

but

*g** (f (x))=g *(*2x+3*)=(*2x+3*)^{2}+(*2x+*3)

=4*x*^{2}+12*x*+9+*2x+*3=4

*x*

^{2}+14

*x*+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.

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