Question

If the domain of the function \( f(x) = \frac{1}{\sqrt{3x + 10 - x^2}} + \frac{1}{\sqrt{x + |x|}} \) is \( (a, b) \), then \( (1 + a)^2 + b^2 \) is equal to:

• A. 25
• B. 16
• C. 24
• D. 26

Hint:

When finding the domain of a function involving square roots, ensure that the expressions inside the square roots are non-negative, and solve the resulting inequalities.

Answer 1

The Correct Option is C

To find the domain of the function \( f(x) \), we need to analyze the restrictions given by the square roots. 1. The term \( \sqrt{3x + 10 - x^2} \) requires the argument inside the square root to be non-negative: \[ 3x + 10 - x^2 \geq 0 \] This is a quadratic inequality. Solving \( 3x + 10 - x^2 = 0 \) by factoring: \[ x^2 - 3x - 10 = 0 \quad \Rightarrow \quad (x - 5)(x + 2) = 0 \] So the values of \( x \) must lie between \( -2 \) and \( 5 \), i.e., \( -2 \leq x \leq 5 \). 2. The term \( \sqrt{x + |x|} \) requires the argument inside the square root to be non-negative. - For \( x \geq 0 \), \( |x| = x \), so \( \sqrt{x + x} = \sqrt{2x} \), which is valid for \( x \geq 0 \). - For \( x<0 \), \( |x| = -x \), so \( \sqrt{x - x} = \sqrt{0} \), which is valid only at \( x = 0 \). Thus, combining these two conditions, the domain of \( f(x) \) is \( [0, 5] \). Therefore, the domain is \( (a, b) = (0, 5) \). Now, we calculate \( (1 + a)^2 + b^2 \): \[ (1 + 0)^2 + 5^2 = 1^2 + 25 = 1 + 25 = 26 \] Thus, the correct answer is \( 26 \), and the correct option is (4).