More on Greatest Integer Function

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We have understand some properties and graph of greatest integer function on my previous post [ click here]


Understand this,

 [2] = 2
[2.09] = 2
[2.8] = 2
[2.98] = 2
[3] = 3

 i.e. for  2 \le x < 3, [x] = 2
it can also see from the graph for 2 \le x < 3, y = 2

 1. Consider,

 [x] = 5

 what is the solution of this equation ??

 [x] = 5 => 4 \le x < 5 or x \in [4,5)

 and, [x] = -3 
then ,  - 3 \le x < - 2 or x \in [ - 3, - 2)

 most funny one,

 [x] = 0
just don't think x = 0

 if [x] = 0,
then 0 \le x < 1 or x \in [0,1)

 So, if [x] = I and I is an integer,
then I \le x < I + 1 or x \in [I,I + 1)

 2.Consider,

 [x] = 5.6 (No Solution)

 why???

 because greatest integer function gives only integer values.

 then there is no solution and for no value of x the above equation satisfied.

 so if [x] = n and n is not an integer,
then there is no solution.

 3. Consider,
[x] > 5
we know that,
for 5 \le x < 6 or x \in [5,6), [x] = 5

 so if [x] > 5, then x \ge 6

 in general,[x] > I, then x \ge I + 1 where I is an integer  

 4. Consider,
[x] > 5.3

 we know that, 
for 5 \le x < 6 or x \in [5,6), [x] = 5

 so if [x] > 5.3, then x \ge 6

 in general if [x] > n, then x \ge I + 1 where I is an integer and [n] = I
5. [x] \ge 5

 we know that
for 5 \le x < 6 or x \in [5,6), [x] = 5

 Since [x] can be equal to 5 
therefore, x \ge 5

 in general, if [x] \ge I, then x \ge I where I is an integer

 6. [x] < 5

 since, for 5 \le x < 6 or x \in [5,6), [x] = 5
therefore, x < 5

 so, if [x] < I, then x < I, where I is an integer

 7. [x] \le 5

we know that
for 5 \le x < 6 or x \in [5,6), [x] = 5

 therefore, x < 6

 so, if [x] \le I, then x < I + 1, where I is an integer

 SO THE IMPORTANT RESULTS ARE : (UNDERSTAND IT )

 1, If [x] = I and I is an integer,
then I \le x < I + 1 or x \in [I,I + 1)

 2.  If [x] = n and n is not an integer,
then there is no solution.

 3. If [x] > I, then x \ge I + 1 where I is an integer  

 4. If [x] \ge I, then x \ge I where I is an integer

 5. If [x] < I, then x < I, where I is an integer

 6. If [x] \le I, then x < I + 1, where I is an integer









Comments

  1. Unknown8 December 2017 at 02:36

    If [x+1/2]>0 then x?

    ReplyDelete
    Replies
    1. abhijatindia8 December 2017 at 13:02

      See the property 3:

      That is if [x] > I => x >= I + 1.

      Therefore, when [x + 1/2] > 0
      Implies x + 1/2 >= 0+1

      Therefore, x >= 1-1/2

      Or, x >= 1/2

      Delete

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