FUNCTIONS

INTRODUCTION

Often, we use word function in every-day life, but we rarely think about its mathematical meaning.

For example, Mary wants to buy lipstick. She is searching for modern, not too expensive one. Each lipstick has its own price. We can say that we have a set of lipsticks, a set of prices and a rule according to which is every element from the second set assigned to the one and only one element of the first set. A price is assigned to each lipstick. We have a function, which assigns the price to the lipstick.

Definition: Suppose that to each element in a set E is assigned a unique element of a set F. Such an assignment is called a function. It is denoted by f .

f : E F

We usually denote functions with small letters, i.e. f, g, h,… Function  f   assigns the element f(x) F to the element x E. Function f of E into F is called a mapping of E into F.

The set E is called the domain of the function  f and F is called the co-domain of f . The element in F, which is assigned to x E is called the image of x and is denoted by Im f .

Element y F is in Im f   if and only if there exists at least one x  E such that f(x)= y. We say that this x is original of y.

Functions can be given in various ways: by the use of tables, formulas, graphs, etc. If the domain of the function is finite set, then we can define such a function in a table. In the first row of the table, we write the elements of the domain, since the second row contains appropriate values of its codomain.

Defining the function with a formula is the most usable way of defining a function in mathematics.

In every day life, especially in scientific research, different measurement equipment is often used, for example oscilloscopes. While measuring electric pulses, different curves are drawn.

If we denote set of real numbers with R, then  f : R R that is defined with f (x) = x²  , has a range consisted of positive real numbers and zero.

Let N be the set of all natural numbers. If  f : N N is defined with f (x)=2x, x N; then f(4) = 2 × 4 = 8, f (11) = 2 × 11 = 22  

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CONSTANT FUNCTIONS

A function f of E into F is called a constant function if the same element f (x) F is assigned to every element in E.  f : E F is a constant function if the range of  f consists of only one element.

Example 1. Let the function  f be defined by the diagram: a,b,c E; 1,2,3 F

Then f   is a constant function since 1 is assigned to every element in E.

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IDENTITY FUNCTION

Let E be any set. Let the function  f : E E be defined by the formula f(x) = x, i.e. let  f assign to each element in E the element itself. Then  f is called the identity function or the identity transformation on E. It is denoted by 1_E.

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EQUAL FUNCTIONS

We say that two functions  f and g are equal and we write f = g , if and only if  the following three conditions are fulfilled:

Functions f and  g are defined on the same set D and have same domains,

Functions f and  g have same co-domains

and f(a ) = g(a ) for every a D.

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RESTRICTIONS AND EXTENSIONS OF FUNCTIONS

Let f be a function of E into G, i.e. let f : E G, and let F be a subset of E. Then  f induces a function f ’ : E G which is denoted by f ’(x) = f (x) for any x F. The function f ’ is called the restriction of f to x and is denoted by f | F.

Let f be a function of E into G, i.e. let f : E G, and let F be  a superset of E. Then the extension of  f  is a function f * : F G which is denoted by f *(x) = f (x), for every x E.

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SET FUNCTIONS

Let f be a function of E into F and let A be a subset of E, that is,  f : E F and A E.

Then f (A) is defined to be the set of image points of elements in A. In other words,

f(A) = { x | f (a) = x, aA, x F}, f(A) F.

A function is called a set function if its domain consists of sets.

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PRODUCT FUNCTION

We can get a new function by the use of two already known functions.

Consider that E, F and G are three non-empty sets. Let f be a function of E into F and let g be a function of F (the co-domain of f) into G. Let x E. Then f(x) F is the domain of g.

Let f(x) F; then its image g(f(x)) G. There is a function of E into G, which is called the product function or the composition function of f and g and it is denoted by (g  f) or (gf). If f : E F and g: F G then (g  f) : E F is defined with (g  f)(a) ekvivalentno g(f(a)).

Example: Let N be set of all natural numbers and f, g two functions

f : N N and g : N N such that

f(n)=n² and g(n)=n+1, for each n N.

Then h = g   f, h : N N

h(n)=n²+1, for each n N

because h(n) = g (f (n) ) = g (n²) = n² + 1.

Products of functions is associative

Let f : A B, g : B C and h : C D. Then we can form the product function g  f: A C, and then the function h (g  f): A D.

Theorem: Let f : A B, g : B C and h : C D then (h g)  f = h (g  f). Then we can write h g f : A D.

Functions have their diagrams. The symbol A   ^f  B denotes a function of A into B.

The diagram

consists of letters A,B,C denoting sets, arrows f,g,h denoting functions f : A B, g : B C and h : A C, and the sequence of arrows {f,g} denoting the composite function g    f   : A C. Each of the functions h : A C and g    f   : A C, that is, each arrow or sequence of arrows connecting A and C is called a path from A to C.

Definition: A diagram of functions is said to be commutative if for any sets X and Y in the diagram, any two paths from X to Y are equal.

Example:

A = {a, b, c}  

B = {x, y, z}

f(a) = x,     f(b) = y,    f(c) = z;

g(x) = a,    g(y) = b,    g(z) = c

We have

(g  f)(a) = a,    (g f)(b) = b,    (g  f)(c) = c

(f  g)(x) = x,    (f  g)(y) = y,    (f  g)(z) = z

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SURJECTION

Consider that E and F are two non-empty sets. Let f : E F.  If f maps E into F then there is no need for each element in F to be an image of the element in E. If the image of function f : E F is set F, we call f the surjection or   the range of f. More precisely, the range of f consists of those elements in F which appear as the image of at least one element in E. We denote it by f(E). The range of the function f has only those elements in F, which represent the image of at least one element in E. It is obvious that f(E) F.

There are no not-assigned elements in the codomain, hence two or more elements of the domain can be assigned to the same element of the codomain.

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ONE-ONE FUNCTION (Injection)

Let f : E F be a function that assigns different elements in E to different elements in F. Then f is called a one-one function (injection). In such a case there are no two different elements in E that have the same image.

f : E F is one-one if f(x)=f(y) x = y            (x,y) E
or
f : E F is one-one if x y implies f(x) f(y), (x,y) E

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BIJECTION

Let f be a function of E into F. Then f(E) F. If every element in F is an image of at least one element in E, then f is a function of E onto F. Then f(E)= F. Bijection is at the same time injection and surjection.

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INVERSE OF A FUNCTION

Let f be a function of E into F, and let y F. Then the inverse of y is denoted by f^-1(y). It consists of those elements in E which are mapped onto F. If f : E F then f^-1(y) = {x | x E, f(x)=y}. f^-1(y) is always a subset of E.

 

INVERSE FUNCTION

Let f be a function of E into F. f^-1(y) could consist of more than one element or might even be the empty set . If f : E F is an one-one function and an onto function, then, for each y F, the inverse f^-1(y) will consist of a single element in E. Each y F has a unique element f^-1(y) E. f^-1 is a  function of F into E. f^-1: F E. In this situation, when f : E F is one-one and onto, we call f^-1 the inverse function of f.

Theorem: Let f : E F be bijection. Then exists one and only one bijection g : F E  that implies

1 (g  f)(x)=x             x E

2 (f g)(y)=y             y F

That unique bijection g is called the inverse function of f.

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