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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Question
8
![**Find the Domain of the Following Function**
\[ f(x) = \frac{5}{81 - x^2} \]
**The domain is:**
\[ \_\_\_\_\_\_\_ \]
*(Type your answer in interval notation.)*
---
**Explanation:**
In this problem, to find the domain of the function \( f(x) = \frac{5}{81 - x^2} \), one must determine the values of \( x \) for which the function is defined. This involves identifying any values of \( x \) that would cause the denominator to become zero, making the function undefined.
To find these values, set the denominator equal to zero and solve for \( x \):
\[ 81 - x^2 = 0 \]
\[ x^2 = 81 \]
\[ x = \pm 9 \]
The function is undefined at \( x = 9 \) and \( x = -9 \).
Therefore, the domain of the function is all real numbers except \( x = 9 \) and \( x = -9 \). In interval notation, this is:
\[ (-\infty, -9) \cup (-9, 9) \cup (9, \infty) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11a64bba-74c0-4fd7-b16b-3d8fff0a6ac2%2F8f7c3aa0-6811-4943-9601-86b6e40dbb4d%2Fweinoup_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find the Domain of the Following Function**
\[ f(x) = \frac{5}{81 - x^2} \]
**The domain is:**
\[ \_\_\_\_\_\_\_ \]
*(Type your answer in interval notation.)*
---
**Explanation:**
In this problem, to find the domain of the function \( f(x) = \frac{5}{81 - x^2} \), one must determine the values of \( x \) for which the function is defined. This involves identifying any values of \( x \) that would cause the denominator to become zero, making the function undefined.
To find these values, set the denominator equal to zero and solve for \( x \):
\[ 81 - x^2 = 0 \]
\[ x^2 = 81 \]
\[ x = \pm 9 \]
The function is undefined at \( x = 9 \) and \( x = -9 \).
Therefore, the domain of the function is all real numbers except \( x = 9 \) and \( x = -9 \). In interval notation, this is:
\[ (-\infty, -9) \cup (-9, 9) \cup (9, \infty) \]
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