FUNCTIONS

Function
Function is a special type of relation. It is one of the most important concepts in mathematics.The word function is derived from a Latin word meaning operation and the word mapping and map are synonimous to it. Functions play very important role in Calculus.

• A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.
• In other words, a function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f  have the same first element.

If f is a function from from A to B and ( a, b ) ∈ f, then f ( a )  = b, where b is called the image of a under f and a is called the preimage of b under f.

• The function f from A to B is denoted by f : A ⟶ B.
• A function which has either R ( real numbers ) or one of its subsets as its range is called a real valued function.
• Further, if its domain is also either R ( real numbers ) or subset of R, it is called real function.

Example

• Let A = { -2, -1, 0, 1, 2 } and B = { 0, 1, 2, 3, 4, 5, 6 } 
• f  be a function, f : A ⟶ B given by f ( x ) = x²
  then f ( -2 ) = 4, f ( -1 ) = 1, f ( 0 ) = 0, f ( 1 ) = 1, f ( 2 ) = 4.
• Here domain ( f ) = { -2, -1, 0, 1, 2 } = A and range ( f ) = { 0, 1, 4 }.

Different types of functions

• Identity function
• constant function
• Polynomial function
• Rational function
• Modulus function
• Signum function
• Greatest integer function

Algebra of real functions

• Addition of two real valued functions:  Let f : X ⟶ R and g : X ⟶ R be any two real functions where X ⊂ R. Then we define  ( f + g) : X ⟶ R by ( f + g ) ( x ) = f ( x ) + g ( x ), for all x ∈ X.
• Subtraction of two real valued functions:  Let f : X ⟶ R and g : X ⟶ R be any two real functions where X ⊂ R. Then we define  ( f - g ) : X ⟶ R by ( f - g ) ( x ) = f ( x ) - g ( x ), for all x ∈ X.
• Multiplication by a scalar: Let f : X ⟶ R be a real valued function and α f is a function from X to R defined by ( α f ) ( x ) = α f ( x ), x ∈ X.
• Multiplication of two real functions: The product of two real functions f : X ⟶ R and g : X ⟶ R is a function f g : X ⟶ R defined by ( f g ) ( x ) = f ( x ) g ( x ), for all x ∈ X. This is also called pointwise multiplication.
• Quotient of two real functions: Let f and g be two real functions defined from X ⟶ R where X ⊂ R. The quotient of f by g denoted by f / g is a function defined by ( f / g ) ( x ) = f ( x ) / g ( x ), provided g ( x ) ≠ 0, x ∈ X.