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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, 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 : 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)=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.
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