Question: Domain of f(x) = sin<sup>−1</sup>$\left( \frac{\lbrack x\rbrack}{\{ x\}} \right)$, where [.] and {...

Question

Domain of f(x) = sin⁻¹ $\left( \frac{\lbrack x\rbrack}{\{ x\}} \right)$ , where [.] and {.} denote the greatest integer function and fractional part respectively, is

Answer 1

Solution

We must have, −1 ≤ $\frac { [ x ] } { \{ x \} }$ ≤ 1

From −1 ≤ $\frac { [ x ] } { \{ x \} }$ , we get $\frac { \{ x \} + [ x ] } { \{ x \} }$ ≥ 0

⇒ $\frac { x } { \{ x \} }$ > 0 ⇒ x ∈ (0, ∞) ~ I⁺ .

From, $\frac { [ x ] } { \{ x \} }$ ≤ 1 we get, $\frac { [ x ] - \{ x \} } { \{ x \} }$ ≤ 0

⇒ [x] ≤ {0} ⇒ x ∈ (-∞, 0)∪(0, 1)~I⁺

Thus domain is (0, 1).

Question: Domain of f(x) = sin<sup>−1</sup>\(\left( \frac{\lbrack x\rbrack}{\{ x\}} \right)\), where [.] and {...

Solution