**Problem Statement:**Find the domain of the function \( f(x) = \sqrt{\frac{30x^3 + 31x^2 - 25x - 6}{12x^3 - 59x^2 - 11x + 30}} \).**Explanation:**To determine the domain of the function \( f(x) \), we need to consider two conditions imposed by the square root and the rational function:1. **Numerator and Denominator**: - The expression inside the square root (the entire fraction) must be greater than or equal to zero: \[ \frac{30x^3 + 31x^2 - 25x - 6}{12x^3 - 59x^2 - 11x + 30} \geq 0 \] - The denominator must not be zero: \[ 12x^3 - 59x^2 - 11x + 30 \neq 0 \]**Finding the Domain:**- **Inequality Analysis**: Solve the inequality \( 30x^3 + 31x^2 - 25x - 6 \geq 0 \) and \( 12x^3 - 59x^2 - 11x + 30 > 0 \) to find intervals where the expression is non-negative and the denominator is not zero. - **Critical Points Identification**: Determine the roots of both the numerator and the denominator by solving \(30x^3 + 31x^2 - 25x - 6 = 0\) and \(12x^3 - 59x^2 - 11x + 30 = 0\) to find critical points that may change the sign of the function.- **Sign Analysis**: Use a sign chart to check on which intervals the expression is positive, considering the critical points.**Conclusion**: The domain will be the union of intervals satisfying both conditions. Ensure to exclude points where the denominator is zero.By identifying and analyzing the roots of the cubic equations, you will determine the set of real numbers \( x \) where the original function is defined.

Name: **Problem Statement:**Find the domain of the function \( f(x) = \sqrt
Uploaded: 2024-03-20T07:07:26.000Z
Channel: Bartleby
Description: **Problem Statement:**Find the domain of the function \( f(x) = \sqrt{\frac{30x^3 + 31x^2 - 25x - 6}{12x^3 - 59x^2 - 11x + 30}} \).**Explanation:**To determine the domain of the function \( f(x) \), we need to consider two conditions imposed by the