Theory of Functions
Function is not a very dangerous topic in mathematics but only a logical approach of relations. If you are week in functions it mean there is something wrong with your basic mathematics. The following topic will help you to know the basic concepts and revise your studies of function.
FUNCTION is nothing but a relation between two sets. Sets are collection of well-defined objects or elements. In this relation the first set is called ?Domain? and second set is called co-domain and the elements which are connected from second to first set are make range.
Generally if there is no specific Domain & Co-domain is defined then Domain and Co-domain are sets of real numbers (As in case of calculus). Finding the domain of a function means finding the subset of (or interval of) real numbers where the function has a real value. To find the range of the function put f(x) equal to y than solve the equation for x as x= g(y) form than find the values where g(y) is not real, by removing these values from real number set we get the range. (Fig. 1)
As in the case of IIT aspirant, we have a general relation as shown above. These type of relations are called function if they fulfill two necessary conditions:
1. Each element in set A should be related to set B (it may be possible some elements of set B are not related to elements of A).
If xÎA there must be a element in B y which is related to A, y=f(x).
2. Each element of set A should be related with only one element of set B. (It may be possible that two elements of set A are related to a single element of set B).
If a is related to b1 and a is related to b2 than b1= b2. But a1 and a2 can be related to b for example in the function y=x2. 2 and ?2 are related to 4. (fig2)
Classical Definition: If two variables x and y are so related that for every value of x, there is a definite value of y, then y is said to be a function of x. Unless otherwise stated, it will be assumed that the variable x has real values only and the corresponding value of y are also real sometimes y is written as f(x) which denotes that y is a function of x. Here x is called ?independent variable? and y is called ?dependant variable? (cause the values of y, depend on values of x).
Kind of Function:
(Fig 3)
Function may be classified as :
2. Explicit & Implicit Functions:
Explicit: Can be expressed as y=f(x). Example y=x2, y=sin x
Implicit: Can?t be expressed as y=f(x). Example x3 + y3 = 3axy, in this function we cannot separate all y in left hand side and x in right hand side.
3. Single & Multi-valued Functions:
Single valued or one-to-one (INJECTION) : Each dependant variable (y) is related to only one independent variable(x). Ex y= 3x.
Multi-valued: Each dependant variable (y) is related to more than one independent variable (x). Ex. y=x2
4. Injection, Surjection, Bijection:
Injection: A function f: A ® B is said to be injection if different elements of A have different images in B (one ? to ?one, two elements of A are not related to single y). Number of one-to-one functions: nPm (only if n>=m where m is no of elements in A and n is no of elements in A).
Surjection: A function f: A ® B is said to be a surjection function if all elements of B have pre-images in A (onto, no element in B has left).
Co-domain = Range. Number of onto functions:
Bijection: A function f: A ® B is said to be bijection if it is one ? to- one and onto. Number of bijections n! (for bijection m = n ).
5. Odd and Even Functions:
Odd Function: A function of f(x) is called an odd function if f(-x) = -f(x). For example y=x3. for x=1 y=1 but for x= -1 y= -1.
Even Function: A function is called even if we replace x with ?x then there is no change in y (no distinct y). Ex. Y=X2.
6. Continuous and Discontinuous Function:
Continuous Functions: A function y=f(x) is said to be continuous in the given interval [a, b] if for every value of independent variable x in the given interval the dependant variable y assume a finite value. (In general if in the given interval we trace a graph of that function we need not to lift our pencil from the paper ? Unbroken Curve). To check the continuity we use limits. Left Hand Side limit = Right Hand Side Limit = Value at the point.
Discontinuous Function: If a function is not continuous in the given interval at any point than it is called discontinuous function. There are three types of discontinuities:
(a) Removable Discontinuity: A function f is called to be has removable discontinuity at x=a if and is not equal to
(b) Discontinuity of I kind: A function f has a discontinuity of I kind at x=a if and both exist and are not equal. It may be from left when or from right when
(c) Discontinuity of II kind: A function f has a discontinuity of II kind at x=a if either or both does not exist .
7. Periodic Functions: If a function repeats its values after a fix interval (Period) than it is called a periodic function. Mathematically we can write that a function is periodic if there exist a interval (period) such that
for all xÎR.
So main thing to remember in function is ? Meticulously check the movement in y as x changes it value. The graphical presentations of functions are very important to study.
(There is some error in this file especially in point 6 & 7 Please see the attached files for clarity and all figures and formula)
Moderation Team
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