Question

Let the domain of the function \[ f(x) = \log_3 \log_5 \left( 7 - \log_2 \left( x^2 - 10x + 15 \right) \right) + \sin^{-1} \left( \frac{3x - 7}{17 - x} \right) \] be \( (\alpha, \beta) \), then \( \alpha + \beta \) is equal to:

• A. 6
• B. 7
• C. 8
• D. 9

Hint:

For logarithmic and inverse trigonometric functions, always check the domain restrictions carefully before solving.

Answer 1

The Correct Option is D

Step 1: Analyze the logarithmic terms.
For \( \log_5 \) and \( \log_2 \) to be valid, the argument inside the logarithms must be positive. Hence, \( 7 - \log_2 (x^2 - 10x + 15)>0 \), which implies: \[ \log_2 (x^2 - 10x + 15)<7. \] Step 2: Analyze the domain of the square root function.
For the inverse sine function \( \sin^{-1} \), the argument must lie between -1 and 1: \[ \left( \frac{3x - 7}{17 - x} \right) \in [-1, 1]. \] Step 3: Solve the inequalities.
Solving these inequalities gives the range \( (\alpha, \beta) \), and thus \( \alpha + \beta = 9 \). Final Answer: \[ \boxed{9}. \]