Question

The domain of the function \( f(x) = \frac{\cos^{-1}x}{[x]} \) is
 

• A. \( [-1, 0) \cup \{1\} \)
• B. \( [-1, 1] \)
• C. \( [-1, 1) \)
• D. None of the above

Hint:

To find the domain of a fraction, find the domain of the numerator and the domain of the denominator, then take their intersection. Finally, explicitly remove any values of \( x \) that make the denominator equal to zero.

Answer 1

The Correct Option is A

To find the domain of the function \( f(x) = \frac{\cos^{-1}x}{[x]} \), we need to consider the conditions for both the numerator and the denominator. 

• Numerator (\( \cos^{-1}x \)): The domain of the inverse cosine function is \( -1 \leq x \leq 1 \). So, \( x \in [-1, 1] \). 
• Denominator (\( [x] \)): The denominator cannot be zero. \( [x] \) is the greatest integer function (floor function). We must find where \( [x] = 0 \). This occurs for all \( x \) such that \( 0 \leq x < 1 \). So we must exclude the interval \( [0, 1) \) from our domain. 

Combining both conditions: We start with the domain from the numerator: \( [-1, 1] \). We must exclude the interval where the denominator is zero: \( [0, 1) \). So, the domain is \( [-1, 1] - [0, 1) \). This can be written as the union of two parts: Part 1: The interval from -1 up to (but not including) 0, which is \( [-1, 0) \). Part 2: The single point at the end of the original interval, which is \( \{1\} \). Therefore, the domain is \( [-1, 0) \cup \{1\} \).