Function as a Special Kind of Relation

Definition: Let A and B  be two non-empty sets. A relation f from A to B , i.e., a sub-set of A \times B , is called a function  from A to B , if
(i) for each a \in A there exists b \in B such that (a,b) \in f
(ii) (a,b) \in f and (a,c) \in f +b=c .

Therefore a non-void subset f of A \times B is a function from A to B if each element of A appears in some ordered pair in f $ and-no two ordered pairs in f have the same first element. If (a,b) \in f then b is called the image of a under f .

Example: Reference – From Wikipedia, the free encyclopedia 

Function as a Correspondence

Definition: Let A and B be two non-empty sets. Then a function f from set A to set B is a rule or method or correspondence which associates elements of set A to elements of set B such that:

i) all elements of set A are associated to elements in set B .
ii) an element of set A is associated to a unique element in set B .

In other words, a function 'f' from a set A to a set B associates each element of A to a unique element of set B .

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Let f :A \rightarrow B be a function such that the set A consists of a finite number of elements.Then, f(x) be described by listing, the values which it attains at different points of its domain.

Domain, Co-domain and Range of a Function

Let f :A \rightarrow B . Then the set A is known as the domain of f and the set B is known as the co-domain of f . The set of all f images of elements of A is known as the range of f or image set of A  under f and is denoted by f (A) .

Thus, f (A) = \{ f(x) : x \in A \} = \ Range \ of \ f

Also f(A) \subseteq B

Example: Let A = \{ -2, -1, 0, 1, 2 \} and B = \{ 0, 1,2, 3,4,5,6 \} . Consider a rule f (x) =x^2 . Under this rule, we obtain f(-2)=(-2)^2 =4, \ \ f(-1)=(-1)^2 =1, \ \ f(0)=0^2 =0, \ \ f (1)=1^2 =1 and f (2) =2^2 = 4 . We observe that each element of A is associated to a unique element of B . So, f : A \rightarrow B given by f(x) = x^2 is a function.

Clearly, domain (f)=A={-2,-1,0,1,2} and range (f)={0,1,4} .

Equal Functions : Two functions f and g are said to be equal if and only if (i) domain of f = domain of g (ii) co-domain of f = co-domain of g , and (iii) f (x) = g(x) for every x belonging to their common domain. If two functions f and g are equal, then we write f = g .

Example: Let A = \{ 1, 2 \} , B = \{ 3, 6 \} and f : A \rightarrow B given by f (x) = x^2 + 2 and g : A \rightarrow B  given by g(x) = 3x . Then, we observe that f and g have the same domain and co-domain. Also we have, f (1)= 3=g(1) and f (2)=6=g(2) . Hence, f=g .

Real Functions: functions having domain and co-domain both as subsets of the set $latex R of all real number. Such functions are called real functions or real valued functions.

Definition: A function f : A \rightarrow B is called a real valued function, if B is a subset of R ( set of all real numbers).

If both A and B both are subsets of R , then f is called a real function.

Domain of a Real Function: the domain of the real function f(x) is the set of all those real numbers for which the expression for f (x) or the formula f(x) assumes real values only.

Range of a Real Function: The range of a real function of a real variable is the set of all real values taken by f (x) at points in its domain.

Standard Real Functions and their Graphs (to be linked)

Operations on Real Functions

Addition: If f : D_1 \rightarrow R and g : D_2 \rightarrow R be two real functions. Then, their sum f + g is defined as that function from D_1 \cap D_2 to R which associates each x \in D_1 \cap D_2 to the number f (x) + g (x) .

(f+g)(x) = f(x) + g(x)   for all x \in D_1 \cap D_2

Product: If f : D_1 \rightarrow R and g : D_2 \rightarrow R be two real functions. Then, their product fg is a function from D_1 \cap D_2 to R and is defined as

(fg) (x) = f(x) g(x)   for all x \in D_1 \cap D_2

Subtraction: If f : D_1 \rightarrow R and g : D_2 \rightarrow R be two real functions. Then, their difference f - g is defined as

(f-g)(x) = f(x) - g(x)   for all x \in D_1 \cap D_2

Quotient: If f : D_1 \rightarrow R and g : D_2 \rightarrow R be two real functions. Then, their quotient \frac{f}{g}   is a function from D_1 \cap D_2  - \{ x:g(x)=0\} to R and is defined as

\Big( \frac{f}{g} \Big) (x) = \frac{f(x)}{g(x)} for all x \in D_1 \cap D_2  - \{ x:g(x)=0\}

Multiplication of a Function by a Scalar: Let f : D \rightarrow R be a real function and \alpha be a scalar (real number). Then the product \alpha f is a function from D to R and is defined as

(\alpha f) (x) = \alpha f (x) for all x \in D

Reciprocal of a Function: If f \colon D \rightarrow R is a real function , then its reciprocal function \frac{1}{f}   is a function from D - \{ x: f(x)  = 0 \} to R and is defined as

\Big( \frac{1}{f} \Big) (x) = \frac{1}{f(x)}