Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Understanding Function Domains
When analyzing functions, one critical aspect is determining their domain—the set of all possible input values (x) that a function can accept without causing mathematical inconsistencies like division by zero or taking the square root of a negative number.
#### Problem 1: Find the Domain
Consider the following functions:
**A)** \[ f(x) = \frac{3x^2 - 27}{|5x - 2| - 3} \]
To determine the domain of this function, we look for values of \( x \) that do not result in division by zero or any other undefined mathematical operations.
**B)** \[ f(x) = \sqrt{\frac{x^2 + 7x + 10}{x + 4}} \]
In this function, we must ensure that the expression inside the square root remains non-negative and the denominator is not zero.
### Steps to Determine the Domain:
**For Function A:**
1. Identify the denominator \( |5x - 2| - 3 \).
2. Determine the values of \( x \) that make the denominator zero, because division by zero is undefined.
3. Set up the equation \( |5x - 2| - 3 = 0 \) and solve for \( x \).
**For Function B:**
1. Ensure that the entire expression inside the square root is non-negative: \( \frac{x^2 + 7x + 10}{x + 4} \geq 0 \).
2. Identify the values of \( x \) that cause the denominator to be zero, as division by zero is undefined.
3. Set up and solve the inequality \( \frac{x^2 + 7x + 10}{x + 4} \geq 0 \), and find the critical points where \( x + 4 = 0 \) and where \( x^2 + 7x + 10 = 0 \).
### Conclusion
Analyzing the domain of these functions involves solving equations and inequalities to identify any restrictions on \( x \). This ensures the function is defined and can produce real number outputs for particular input values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc156cfcf-b572-4dd1-9c92-c49112da643d%2F82750782-3380-4e14-a252-d1fbe741bb13%2Fymi4xoh_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Function Domains
When analyzing functions, one critical aspect is determining their domain—the set of all possible input values (x) that a function can accept without causing mathematical inconsistencies like division by zero or taking the square root of a negative number.
#### Problem 1: Find the Domain
Consider the following functions:
**A)** \[ f(x) = \frac{3x^2 - 27}{|5x - 2| - 3} \]
To determine the domain of this function, we look for values of \( x \) that do not result in division by zero or any other undefined mathematical operations.
**B)** \[ f(x) = \sqrt{\frac{x^2 + 7x + 10}{x + 4}} \]
In this function, we must ensure that the expression inside the square root remains non-negative and the denominator is not zero.
### Steps to Determine the Domain:
**For Function A:**
1. Identify the denominator \( |5x - 2| - 3 \).
2. Determine the values of \( x \) that make the denominator zero, because division by zero is undefined.
3. Set up the equation \( |5x - 2| - 3 = 0 \) and solve for \( x \).
**For Function B:**
1. Ensure that the entire expression inside the square root is non-negative: \( \frac{x^2 + 7x + 10}{x + 4} \geq 0 \).
2. Identify the values of \( x \) that cause the denominator to be zero, as division by zero is undefined.
3. Set up and solve the inequality \( \frac{x^2 + 7x + 10}{x + 4} \geq 0 \), and find the critical points where \( x + 4 = 0 \) and where \( x^2 + 7x + 10 = 0 \).
### Conclusion
Analyzing the domain of these functions involves solving equations and inequalities to identify any restrictions on \( x \). This ensures the function is defined and can produce real number outputs for particular input values.
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