FUNCTIONS

 Definition : 

        A function (or mapping) f from set A to set B (f: A → B ) is a relation which associates for each element x in A , a unique (exactly one) element y in B. Then the element y is expressed as y = f(x). y is the image of x under f.                                           

f is called map or transformation . If such a function exist , then A is called the domain of f and B is called co-domain of f . 

Types of Function :

1.   One-One or One to One or Injective Function :

       A function  f: A → B is said to be one-one if different element in A have different image in B . The condition is aslo expressed as f(a) = f(b) ⇒ a = b [As a ≠ b ⇒f(a) ≠ f(b)] 

2. Onto or Surjective Function :

     A function   f: A → B is said to be onto if every element y in B is an image of some x in A (or y in B haspreimage x in A) .  The image of A can be denoted by f(A) . 

    f(A) = {y ∈ B / y = for some x ∈ A} . 

Note : i)  f: A → B  is anto if f(A) = f . Also range of f = f(A) ⊂ co-domain of f .

    ii) If the function  f: A → B id both one-one and onto then the function is callled bijective function .

Graph of Function 

       If the domain of function is in R , we can show the function by a graph in XY plane . The graph consisits of points (x, y) where , y = f(x) .

Vertical Line Test :

    A graph represent function of x , only if no vertical line intersect the curve in more than one point.

Horizontal Line :

   If no horizontal line intersect the graph of a function in more than one point , then the function is one-one function. The graph is one-one function as a horizontal line intersect the graph at only one point.

Value of Function :

      f(a) is called the value of function f(x) at x = a .

eg., Evalute f(x) = 2 x2 - 3x +4   at x = 7 and x = 3

solution :   f(x) at x = 7 is f(7)

             f(7) = 2(7)2 -3(7) +4 

                   = 2(49) -21 + 4

                   = 98 - 21+ 4 

                   = 81

at x = 3  

 f(3) = 2(3) 2 - 3(3) +4

       = 3(9) - 9 + 4

      = 27 - 9 + 4

       = 22

Some basic Function :

1.   Constant Function :

       A constant function is a function whose value is the same for every input value .

  eg., graph og f(x) = 2 

                 

Domain : R or (-∞ , ∞)  and Range : {2}.

2.    Identity Function :

            If f : R→R then indetity function is defined by f (x) = x , for every x ∈ R.
    Identity function is given in the graph below .
                 
Domian : R or (-∞ , ∞)  and Rnage : R or (-∞ , ∞) 

3.  Power Function :  f(x) = axn  , n ∈ N 
   This function is a multiple of nth power of x .

       i)  Square Function :  f(x) =  x2 .
                         
Domain : R or (-∞ , ∞)  and Rnage : R or [0 , ∞) 

ii) Cube Function :  f(x) = x2 .
                    
Domain : R or (-∞ , ∞)  and Rnage : R or ( ∞ , ∞) 
  
Properties : The graph of odd power of x (more than 1) looks similiar to cube function eg., x5 , x7 .

4.  Polynomial Function :
   
      f(x) = a0xn +a1 xn-1 +...+an-1x +an is polynomial fnction of degree n , if an  ≠ 0 , and ai s are real .
 i)   Linear Function :
      From : f(x) = ax +b (a≠0)
     eg., f(x) = -2x +3 , x ∈ R.
                            
Domain : R or (-∞ , ∞)  and Rnage : R or ( -∞ , ∞) .

ii)   Quadratic Function :
      f(x) = a x2 + bx +c (c≠0 )
                         
Domain : R or (-∞ , ∞)  and Rnage : R or [k , ∞) .

iii)   Cubic Function :  f(x) = a x3  +b x2 + cx + d   (a≠0)

Domain : R or (-∞ , ∞)  and Rnage : R or ( -∞ , ∞) .

5.  Rdical Function :
     
    i)  Square root function :
             In square root of negative number is not a real number , so the domain of √x is restricted to positive value of x .
         f(x) = √x , x ≥0
                      
Domain : R or [0 , ∞)  and Rnage : R or [0 , ∞) .

ii)   Cube root function :

Domain; R and  Range : R 

6.  Rational Function :
     Given polynomials p(x) , q(x) = p(x)/q(x)  is defined for x if q(x) ≠0 .
eg., f(x) = 1/x , x ≠ 0.
Domain : R -{0}   and Range : R -{0}

7.   Exponential Function : 
    f(x) = ax is an exponential function with base a and exponent x , a≠ 0 , a > 0 and x ∈R.
  eg., f(x) =  2x and f(x) = 2-x


Domain : R and Range : (0,∞)


8. Logarithmic Function :
   Let a< 0 , a ≠1, then logarithmic function logax, y =  logax if x =  ay
 for x > 0 , is defined as 
   


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