Functions

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Functions

Maths could be defined as the lore of structures on sets, and one of  the most important mathematical structures are the functions. Other names for ‘function’ are ‘map‘, ‘mapping‘ and ‘transformation‘.

First a brief look at what we have seen on the previous few pages.
In general we call a logical statement about variables (names or ‘pointers’ to objects) a relation. A relation can be seen as an expression that may be true or false, and in the cases it is true we say that the values of variables satisfies the relation. To find which values satisfies the relation is called to solve the relation,  and each possible set of values that satisfies the relation is called a solution to the relation. An example of relation is ‘x is father to y‘ that is satisfied by each father and all his kids. If Stig is father to Anna, then x=’Stig’, y=’Anna’ is a solution to the relation.

Definition of Functions and related terms

If we, for each element in a set A, can assign or associate one and only one element in a set B, using the relation f, then f is a function from A into B.

We can write this as,

f:A\to B

which can be read as f maps (or takes) A into B.

The relation  f could in general be written y f x or f (x,y), where x is a member of A and y is a mender of B.  For the relation to be a function it must be true for one and only one y for each x. We could write this as,

function(f)=\forall x\in A,y\in B,z\in B(f(x,y)\wedge f(x,z)\Leftrightarrow y=z)

This is read ‘ f is function if it is a relation such that for all x in A and all y and z in B we have that if f(x,y) and f(x,z) holds, then y=z‘. In other words, each x correspond to one and only one y.

If we can write the relation in the form y=f (x), then the functions is said to be explicit, or explicitly defined. Not all functions is possible to write in this form though. If it is written in any other form,  f(x,y), then the function f is said to be implicit, or implicitly defined .

The set A is called the domain of f . The set of values y may take in B as x “varies” trough all values of A is called the  range of  f.

The range is in some books defined as any set large enough to cover what we defined as the range. A set like this is usually called the codomain.

The value y is called the image of  x under f . If it is possible to write the function in an explicit form then the image of x under y=f (x) will be the value of f (x).

The symbols x and y are called variables, and x is said to be an independent variable, and y the dependent variable, this because, as commonly seen, when  we choose a value of x, we get the value of y – y is then dependent on x. I would though rather see as – what value of y makes the relation y=f (x) true for a given x.

More on the f (x) notation

The notation f (x)=x2+x is read “the function f of x is x squared plus x” or just “f of x is x squared plus x“. By f (x)=x2+x we mean that to calculate f (x) we replace each occurrence of x in the right hand side of the expression  f (x)=x2+x by whatever we have put instead of x in the left hand side of the expression.  We have for example that f (5)=52+5=25+5=30.

So why is this notation used instead of just writing y=x2+x? There are several reasons

  • Say we have several functions y=x2+x and y=3x. The the notations f (x)=x2+x and g(x)=3x gives the functions different names.  We may then talk about f (x) and g(x) without having to refer to the definitions every time.
  • We can use it as an abbreviation;  “find f (5)” instead of “find the value of y=x2+x when x=5″.
  • We can start to use functions as variables, i.e. we can talk about functions in general; “Say we have a function f (x) such that it goes through the origin”. We are now thus talking about any function going through the origin.

Examples of functions, and non-functions

The relation y=3x where x and y are real numbers (x,y∈R) is a function. there is one and only one real number value 3x for each real x.

The relation y2=x , x,y∈R is not a function because, for all x but 0, we will have two possible values for y.  We havem for example, that for x=4 we may have y=2 and y=-2.

The relation

y=\int\limits_{a}^{b}{f(x)dx}

is a function if the domain is the three-tuple of  two reals a and b, and f (x) is  a function  integratable  in the range [a,b],  and the range being the real numbers.
Here one of the variables is a function.

The relation

F(x)=\int{f(x)dx}

is not a function since F(x) is not fully determined by integration. Any function F(x)+C makes the relation true.

The relation ‘sound of a letter’ is not a function. The pronunciation of the letter ‘o’ in the word ‘word’ and in the word ‘who’ are not the same.

The relation ‘number of the position of a letter in the English alphabet’ is a function. Each letter has one and only one position. This we could show graphically like this:

I have shown the relation for the first four letters only, mainly because I am lazy.

Up a level : Algebra and Arithmetic
Previous page : Relations represented by directed graphs
Next page : The Graph of a functionLast modified: Jan 8, 2017 @ 15:57