Find the domain of function \(f\left( x \right) = \;\frac{{2.3}}{{\sqrt {x - \left[ x \right]} }}\;\), where [.] represents the greatest integer function.

  1. R - Z
  2. R
  3. (-∞, 0) (0, ∞)
  4. None of these

Answer (Detailed Solution Below)

Option 1 : R - Z
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Detailed Solution

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CONCEPT:

The real function f: R → R defined by f (x) = [x], x ∈R picks the value of the greatest integer less than or equal to x, is called the greatest integer function.

Thus f (x) = [x] = – 1 for – 1 ≤ x < 0

And f (x) = [x] = 0 for 0 ≤ x < 1

[x] = 1 for 1 ≤ x < 2

[x] = 2 for 2 ≤ x < 3 and so on

CALCULATIONS:

Given function is \(f\left( x \right) = \;\frac{{2.3}}{{\sqrt {x - \left[ x \right]} }}\;\)

As we know that \(0 \leqslant x - \left[ x \right] < 1,x \in R\)

For x ∈ Z, x – [x] = 0

So, domain is R – Z 
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