Step function or the greatest integer function

Let x ∈ R. A function f(x) = [x] defined as f(x) = n (n is an integer) such that n ≤ x < n + 1 is called a step function.

For all x such that 0 ≤ x < 1

            f(x) = 0

For all x such that -1 ≤ x < 0

          f(x) = -1

          - 2 ≤ x < 1

           f(x) = -2 etc.,

           Domain of f = R

          Range of f = Z

If f(x) = [x + k] = [x] + k where k is an integer

Example 1:

         Solution set of [x] = -2 is { x | -2 ≤ x < -1}

Example 2:

          f: Z→Z defined by f(x) = [x] is one–one and onto.

          f: R→Z defined by f(x) = [x] is onto but

                                                         not one–one.

         f: Z→R defined by f(x) = [x] is one–one

                                                         but not onto.

          f: R→R defined by f(x) = [x] is neither one–one

                                                         nor onto.

Try this question

1.   Identify this type of function

   1.   

Solution:

1.   Step function