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:
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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.
To determine the domain of the function \( f(x) \), we analyze the constraints imposed by the square root expressions.1. The expression \( \sqrt{3x + 10 - x^2} \) necessitates that the radicand be non-negative: \[ 3x + 10 - x^2 \geq 0 \] To solve this quadratic inequality, we first find the roots of \( 3x + 10 - x^2 = 0 \): \[ x^2 - 3x - 10 = 0 \quad \Rightarrow \quad (x - 5)(x + 2) = 0 \] The roots are \( x = 5 \) and \( x = -2 \). The inequality \( 3x + 10 - x^2 \geq 0 \) holds for \( x \) values between these roots, inclusive: \( -2 \leq x \leq 5 \).2. The expression \( \sqrt{x + |x|} \) also requires a non-negative radicand. - If \( x \geq 0 \), then \( |x| = x \), and the expression becomes \( \sqrt{x + x} = \sqrt{2x} \). This is valid for \( x \geq 0 \). - If \( x<0 \), then \( |x| = -x \), and the expression becomes \( \sqrt{x - x} = \sqrt{0} \). This is only valid at \( x = 0 \).Combining both conditions, the domain of \( f(x) \) is the intersection of \( [-2, 5] \) and \( [0, \infty) \), which is \( [0, 5] \).We are given that the domain is \( (a, b) = (0, 5) \). This implies \( a = 0 \) and \( b = 5 \).Now, we compute \( (1 + a)^2 + b^2 \):\[(1 + 0)^2 + 5^2 = 1^2 + 25 = 1 + 25 = 26\]The result is \( 26 \), corresponding to option (4).
