Question

The domain of the function \( f(y) = \frac{\cos^{-1(y - 5)}{\sqrt{25 - y^2}} \) is}

• A. \( (4, 6) \)
• B. \( (-5, 5) \)
• C. \( [4, 5] \)
• D. \( (4, 5) \)

Hint:

When finding the domain of inverse trigonometric functions, consider both the restrictions of the function and the square root.

Answer 1

The Correct Option is C

Step 1: Analyze the domain of the function.
For the function to be defined, we must have the following conditions: - \( 25 - y^2 \geq 0 \) (the square root must be non-negative). - \( y - 5 \) must lie between -1 and 1 for \( \cos^{-1} \) to be valid. Step 2: Solve for the domain.
We find that the domain of the function is \( [4, 5] \).
Step 3: Conclusion.
The correct answer is (C) \( [4, 5] \).