30x3 + 31x? 25x – 6 - Find the domain of f(x) = 12x3 – 59x2 11x + 30 - -

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**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 the constraints imposed by the square root and the rational function:

1. **Square Root Constraint:**
   - The expression inside the square root must be greater than or equal to zero:
   \[
   \frac{30x^3 + 31x^2 - 25x - 6}{12x^3 - 59x^2 - 11x + 30} \geq 0
   \]

2. **Rational Function Constraint:**
   - The denominator cannot be zero:
   \[
   12x^3 - 59x^2 - 11x + 30 \neq 0
   \]

By solving the above inequalities, we can determine the values of \( x \) for which the function is defined, thus finding the domain of \( f(x) \).
Transcribed Image Text:**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 the constraints imposed by the square root and the rational function: 1. **Square Root Constraint:** - The expression inside the square root must be greater than or equal to zero: \[ \frac{30x^3 + 31x^2 - 25x - 6}{12x^3 - 59x^2 - 11x + 30} \geq 0 \] 2. **Rational Function Constraint:** - The denominator cannot be zero: \[ 12x^3 - 59x^2 - 11x + 30 \neq 0 \] By solving the above inequalities, we can determine the values of \( x \) for which the function is defined, thus finding the domain of \( f(x) \).
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