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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,

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,

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)=*x*^{2}+*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)=*x*^{2}+*x* we mean that to calculate *f *(x) we replace each occurrence of *x* in the right hand side of the expression *f *(x)=*x*^{2}+*x* by whatever we have put instead of *x* in the left hand side of the expression. We have for example that *f *(5)=5^{2}+5=25+5=30.

So why is this notation used instead of just writing *y*=*x*^{2}+*x*? There are several reasons

- Say we have several functions
*y*=*x*^{2}+*x*and*y*=3*x*. The the notations*f*(x)=*x*^{2}+*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*=*x*^{2}+*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*=3*x *where x and y are real numbers (*x*,*y*∈* R*) is a function. there is one and only one real number value 3

*x*for each real

*x*.

The relation *y*^{2}=*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

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

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.

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