Question

For f(x) = [x] , where [x] is the greatest integer function, which of the following is true, for every x ∈ R.

Options

• [x] + 1 = x

• [x] + 1 < x

• [x] + 1 ≤ x

• [x] + 1 > x

Answer 1

[x] + 1 > x

Explanation:

We have,

f(x) = [x]

We know that, x = [x] + {x}

Where [x] is greatest integer function and {x} is fractional part

= {X} = X - (X)

0 ≤ {x} < 1

∴ 0 ≤ x - [x] < 1

⇒ [x] ≤ x < [x] + 1

⇒ [x] + 1 > x