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
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
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