Functions



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Chapter 2 : Functions



Functions Definition arrow_upward


  • A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
  • The output of a function f corresponding to an input x is denoted by f(x) (read " f of x").
  •  

  • Let X and Y be two nonempty sets of real numbers.
  • A function from X into Y is a relation that associates with each element of X a unique element of Y.
  • Domain:
  • The set of all possible input values. For  function f(x), x is called the Domain of the function.
  • Range:
  • The set of all images of the elements of the domain is called the Range of the function.
  • For each element x in X, the corresponding element y in Y is called the image of x.
  • Example of Function:
  • Any sequence can be considered as a function defined on a set of integers.
  • Example: Sequence 2, 5, 8, 11, 14,… can be defined as a function as follows:
  •     f : n → 3n - 1


    One-to-One Functions arrow_upward


  • A one-to-one or injective function f from set X to set Y is a function such that each x in X is related to a different y in Y.
  • F is one-to-one or injective if for all elements  . If  then . No  in the range is the image of more than one  in the domain.
  • As shown in figure that, for every element of  there is exactly one value of.
  • Example: f: R→R is defined by f(n) = 2n + 3, g: R → R is defined by f(x) = x2 , then f is one-to-one function, and g is not.

  • Onto Functions arrow_upward


  • A function f : XY is said to be onto or surjective if for every y in Y, there is an x in X, such that f(x) = y. All the elements in the Y are used.
  • A function is onto if two or more different elements in the domain corresponds to the same element in the range.
  • Examples:
    • We define f: R → R by f(n) = n2 - 2
    • g: R → R by f(n) = 3 n - 4
    • Then, f is onto, and g is not.

    One-to-One Correspondences arrow_upward


  • A One-to-One correspondence or bijection from a set X to a set Y is a function  that is both one-to-one and onto.

  • Independent Variable arrow_upward


  • For a function y = f(x), the variable x is called the Independent Variable.
    • Because it can be assigned any of the permissible numbers from the domain.
  • The independent variable is also called the Argument of the function.

  • Dependent Variable arrow_upward


  • The variable y is called the Dependent Variable, because its value depends on x.

  • Boolean Functions arrow_upward


  • This is a function that returns a true (for 1) or false (for 0) value instead of a numeric value. It uses principal logical relations “and”, “or”, “not”, “if…then” etc.
  • In Boolean function both range and domain consists of true and false.
  • Example:

  • Real Valued Function arrow_upward


  • The function for which the domain and range are the subsets of the set of real number R is called a Real valued Function.
  • Uniform definition
    • If a function is defined as
    • , we say that it is uniformly defined.
  • Piecewise Definition
    • If a function  assumes different forms in different subsets of, we say that it is piecewise defined.

    Algebraic Functions arrow_upward


  • Functions with values that are obtained by adding, subtracting, multiplying or dividing constants and independent variable or raising a variable to a power are known as Algebraic Functions.
  • Algebraic functions are of three types.
  • Polynomial Function:
  • If a function y = f(x) is given by:
  • Where,  are real numbers (sometimes called the coefficients of the polynomial) and n is any non – negative integer, then f(x) is called a Polynomial Function in x.
  • The degree of a polynomial is the highest power of x in its expression. The function f(x) = 0 is also a polynomial but we say that its degree is “undefined”.
  • Example: f(x) = 4x3 – 3x2 + 2 is a polynomial function with degree 3.
  • Rational Functions:
  • If a function  is given by
  • Where  and  are polynomial functions, then f(x) is called a Rational Function in x.
  • Irrational Functions
  • The algebraic functions containing one or more terms having non – integral rational powers of x are called Irrational Functions. 

  • Signum Function arrow_upward


  • Signum function is denoted by ,
  • thus, f (x) = sgn (x)
  •      Where,

         

      

  • Domain of sgn (x) = R
  • Range of sgn (x) = {-1, 0, 1}

  • Exponential Functions arrow_upward


  • If a is any number such that a > 0 and a ≠ 1  then an exponential function is a function in the form, , where a is called the base and x can be any real number.
  • The Exponential Function with base  is the following function from R to .
  •  

     

             


    Logarithmic Functions arrow_upward


  • If a is any number such that a > 0 and a ≠ 1 and x > 0 then, .
  • Symbolically,
  •   

  • Properties
  •     

        

       

      

     


    Inverse Functions arrow_upward


  • Suppose  is a one-to-one correspondence.
  • Then, there is a function  defined as follows:
    • Given any element in Y.   is the unique element x in X such that 

            F(x) = y.

  • The function   is called the Inverse Function for F.
  • Example:
    • The logarithmic function with base  is the inverse of the exponential function with base b.That is logarithmic function y = logb x is the inverse function of exponential function that is by = x.


    Thank You from Kimavi arrow_upward


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