Let `f(x)=cot^(-1)x+ "cos" ec^(-1)x`.Then f(x) is real for

Options

A. `x epsilon[-1,1]`
B. `x epsilon(-infty,-1]uu[1,infty)`
C. `x epsilon(-infty,infty)`
D. None of these

Answer

To find the values of \( x \) for which the function \( f(x) = \cot^{-1} x + \cos^{-1} x \) is real, we need to determine the domains of the individual inverse trigonometric functions involved.

### Step-by-Step Solution:

1. **Determine the Domain of \( \cot^{-1} x \)**:
- The function \( \cot^{-1} x \) is defined for all real numbers \( x \).
- Therefore, the domain of \( \cot^{-1} x \) is:
\[ x \in (-\infty, \infty) \]

2. **Determine the Domain of \( \cos^{-1} x \)**:
- The function \( \cos^{-1} x \) is defined for \( x \) in the interval \( [-1, 1] \).
- Therefore, the domain of \( \cos^{-1} x \) is:
\[ x \in [-1, 1] \]

3. **Combine the Domains**:
- Since \( f(x) \) is the sum of \( \cot^{-1} x \) and \( \cos^{-1} x \), we need to find the intersection of the two domains.
- The domain of \( f(x) \) is the set of \( x \) values that satisfy both conditions:
\[ x \in (-\infty, \infty) \cap [-1, 1] = [-1, 1] \]

4. **Conclusion**:
- The function \( f(x) = \cot^{-1} x + \cos^{-1} x \) is real for:
\[ x \in [-1, 1] \]

### Final Answer:
The function \( f(x) \) is real for \( x \in [-1, 1] \).