Maths-
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Easy

Question

If f colon R not stretchy rightwards arrow R text  is defined by  end text f left parenthesis x right parenthesis equals x minus left square bracket x right square bracket minus 1 half text  fo  end text x where left square bracket x right square bracket is the greatest integer no exeeding x comma text  then  end text open curly brackets x element of R colon f left parenthesis x right parenthesis equals 1 half close curly brackets equals

  1. Z, the set of all integers
  2. N, the set of all natural numbers
  3. straight empty set, the empty set
  4. straight R

Hint:

To find x when f(x)= 1 half for the given function, first we know that x equals open square brackets x close square brackets plus open curly brackets x close curly brackets rightwards double arrow open curly brackets x close curly brackets equals x minus open square brackets x close square brackets, we can write f(x)=open curly brackets x close curly brackets minus 1 half, now will find the condition when f(x) will be 1 half.

The correct answer is: straight empty set, the empty set


    Given, f(x)=x minus open square brackets x close square brackets minus 1 half
    we know that, x=open square brackets x close square brackets plus open curly brackets x close curly brackets
    x minus open square brackets x close square brackets equals open curly brackets x close curly brackets
    Now, f(x) =open curly brackets x close curly brackets minus 1 half
    For, f left parenthesis x right parenthesis equals 1 halfopen curly brackets x close curly brackets minus 1 half equals fraction numerator 1 over denominator 2 space end fraction
open curly brackets x close curly brackets equals 1 space comma space w h i c h space i s space n o t space p o s s i b l e space a s space open curly brackets x close curly brackets space i s space a space f r a c t i o n a l space p a r t space f u n c t i o n comma space 0 less or equal than open curly brackets x close curly brackets less than 1
S o comma f o r space f left parenthesis x right parenthesis equals 1 half comma space space x equals empty set

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