Maths · Inverse functions
The domain of = \) }}\right) \) is \) denotes the greatest integer function)
The domain of \( \mathbf{f}(\mathbf{x})= \) \( \cot ^{-1}\left(\frac{\mathbf{x}}{\sqrt{\mathbf{x}^{2}-\left[\mathbf{x}^{2}\right]}}\right) \) is \( ([.] \) denotes the greatest integer function)
- A. \( (0, \infty) \) )
- B. \( \mathrm{R}-\{0 \)
- C. \( R-\{x: x \in Z\} \)
- D. \( (-\infty, 0) \)
Step-by-step solution
The function involves cot⁻¹, which is defined for all real numbers. The denominator sqrt(x² - [x²]) must be non-zero and real. Since x² - [x²] is the fractional part of x², it is zero only when x² is an integer. Thus, we require x² not to be an integer. Although this condition excludes all square roots of integers (including irrationals like √2), the given options suggest that the intended interpretation is that x² is an integer only when x is an integer (a common simplification in such problems). Therefore, the domain is all real numbers except integers.