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Question

Let Z denote the set of all integers and f colon Z not stretchy rightwards arrow Z defined by f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign center center columnspacing 1em end attributes row cell x divided by 2 end cell cell left parenthesis x text  is even  end text right parenthesis end cell row 0 cell left parenthesis x text  is odd  end text right parenthesis end cell end table text  Then  end text f closeis

  1. onto but not one-one
  2. one-one but onto
  3. one-one and onto
  4. neither one-one nor onto

Hint:

We are given a function f(x). The domain and co-domain of the function are set of all integers. We have to find if the given function is one-one and onto.

The correct answer is: onto but not one-one


    The given function is
    f(x) = x over 2            if x is even
    = 0                if x is odd
    The value of function will be x over 2 when the we substitute even integers for x. When we substitute odd Integers in place of x we get the value 0.
    We have to check if the given function is one-one or/and onto.
    1) One-One function: A function is said to be one-one function if for every element in domain there is unique element in co-domain.
    For this function, every odd number has same image in the co-domain.
    So, the function is not one-one.
    2) onto function: A function is said to be onto if every element in co-domain have some pre image.  It means the domain and range of the function should be equal.
    For this function, co-domain is set of all integers.
    Let's find the range.
    For odd values of x  f(x) = 0
    For even values of x  f(x) = x over 2
    So, if x = 2           f(x) = 2 over 2 = 1
    If x = 4            f(x) = 4 over 2 = 2
    If x = -2            f(x) = fraction numerator negative 2 over denominator 2 end fraction = -1
    If x = - 4         f(x) = fraction numerator negative 4 over denominator 2 end fraction = -2
    So, range ={...-2, -1, 0, 1, 2, ...}
    Range is set of all integers.
    Hence co-domain and range are equal.
    So, the given function is onto.
    The given function is onto but not one-one.

    The letter Z represents set of Integers. We should know definition of different types of functions to solve such questions.

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