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 \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3673ef98-a23e-49c9-abd6-672ebae22fc7%2F5feb971d-7e80-4770-a61d-7e7a37b6df42%2Flhs37z_processed.jpeg&w=3840&q=75)
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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