Question

The domain of the function \[ f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}} \] is \( (a, b) \). Then \( (1 + a^2) + b^2 \) is:

• A. 26
• B. 30
• C. 25
• D. 29

Hint:

When finding the domain of a function with square roots, ensure that the expression inside the square roots is non-negative.

Answer 1

The Correct Option is C

- The first term \( \frac{1}{\sqrt{10 + 3x - x^2}} \) is defined when the expression under the square root is non-negative. The discriminant of \( 10 + 3x - x^2 \geq 0 \) gives the range of \( x \) for which the expression is valid. - The second term \( \frac{1}{\sqrt{x + |x|}} \) requires \( x + |x| \geq 0 \), which is valid for \( x \geq 0 \). By solving these, the domain of the function is \( (a, b) = ( -1, 2 ) \). Now, compute \( (1 + a^2) + b^2 \): \[ (1 + (-1)^2) + 2^2 = 2 + 4 = 6 \] Thus, the correct answer is 6.