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Function Theory
Tags: CBSE Board  |  Differential Calculus  |  Class 12  |  ICSE Class 12  |  Functions and Graphs
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Introduction

1. General Definition : If to every value (Considered as real unless other-wise stated) of a variable x, which belongs to some collection (Set) A, there corresponds one and only one finite value of the quantity y, then y is said to be a function (Single valued) of x or a dependent variable defined on the set A; x is the argument or independent variable.

If to every value of x belonging to some set A there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on A. Conventionally the word ''Function'' is used only as the meaning of a single valued function, if not otherwise stated. Pictorially: y is called the image of x and x is the pre-image of y under f.
Every function from satisfies the following conditions.
a)
b) and
c)

2. Domain, Co-Domain & Range Of a Function : Let , then the set A is known as the domain of f and the set B is known as co-domain of f. The set of all ''f'' images of elements of A is known as the range of f. Thus :
Domain of
Range of

It should be noted that range is a subset of co-domain. Sometimes if only f(x) is given then domain is set of those values of ''x'' for which f(x) exists or is defined.
To find the range of a function, there is n''t any particular approach, but student will find one of these approaches useful.
i) When a function is given in the form y = f(x), express if possible ''x'' as a function of ''y'' i.e. x = g(y). Find the domain of ''g''. This will become range of ''f''.
ii) If y = f(x) is a continuous or piece-wise continuous function, then range of ''f'' will be union of
[Minmf(x), Maxmf(x)] in all such intervals where f(x) is continuous/piece-wise continuous.

3. Classification of Functions :
Functions can be classified into two categories :
i) One-One Function (Injective mapping) or Many - One Function : A function is said to be a one-one function or injective mapping if different elements of A have different f images in B. Thus for .

Diagramatically an injective mapping can be shown as

OR
Note : (a) Any function which is entirely increasing or decreasing in whole domain, then f(x) is one-one.
(b) If any line parallel to x-axis cuts the graph of the function atmost at one point, then the function is one-one.
Many - One Function : A function is said to be a many one function if two or more elements of A have the same f image in B. Thus is many one if for :
but

Diagramatically a many one mapping can be shown as

OR

Note : (a) Any continuous function which has atleast one local maximum or local minimum, then f(x) is many-one. In other words, if there is even a single line parallel to x- axis cuts he graph of the function atleast at two points, then f is many - one.
(b) If a function is one-one, it cannot be many-one and vice versa.
(c) All functions can be categorized as one-one or many-one

ii) Onto function (Surjective mapping) or into function :
If the function is such that each element in B (co-domain) must have atleast one pre-image in A, then we say that f is a function of A ''onto'' B. Thus is surjective iff some such that f(a) = b.
Diagramatically surjective mapping can be shown as

OR
Note that : If range = Co-domain, then f(x) is onto.

Into Function : If is such that there exists atleast one element in co-domain which is not the image of any element in domain, then f(x) is into.
Diagramatically into function can be shown as

OR

Note that :
If a function is onto, it cannot be into and vice versa.
Thus a function can be one of these four types :
a) one-one onto (injective and surjective)
b) one-one into (injective and surjective)
c) many - one onto (surjective but not injective)
d) many-one into (neither surjective nor injective)
( domain in each case is )
Note : a) If f is both injective and surjective, then it is called a Bijective mapping. The bijective functions are also named as invertible, non-singular or biuniform functions.
b) If a set A contains n distinct elements then the number of different functions defined from is nnand out of it n! are one one.

4. Algebraic Operations On Functions : If f& g are real valued functions of x with domain set A, B respectively, then both f and g are defined in . Now we define f + g, f - g, (f.g) and (f/g) as follows:
i)
ii) (f.g) (x) = f(x). g(x)
iii) domain is

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