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...
Related questions
Question
![**Find the domain of the function. (Enter your answer using interval notation.)**
\[ f(u) = \frac{u + 2}{2 + \frac{2}{u + 2}} \]
---
### Explanation
To find the domain of the function, we need to determine for which values of \( u \) the function is defined. The function is a fraction, and it is undefined when the denominator equals zero.
1. **Denominator Analysis:**
\[
2 + \frac{2}{u + 2} \neq 0
\]
- First, identify where \( \frac{2}{u + 2} \) is undefined:
\[
u + 2 \neq 0 \quad \Rightarrow \quad u \neq -2
\]
- Next, solve for when the entire denominator is zero:
\[
2 + \frac{2}{u + 2} = 0
\]
\[
\frac{2}{u + 2} = -2
\]
\[
2 = -2(u + 2)
\]
\[
2 = -2u - 4
\]
\[
6 = -2u
\]
\[
u = -3
\]
2. **Domain:**
- Therefore, the domain is all real numbers except \( u = -2 \) and \( u = -3 \).
### Domain in Interval Notation
\[
(-\infty, -3) \cup (-3, -2) \cup (-2, \infty)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9a46c7d-2d16-4dcd-9d7f-ae6502f786e7%2F79d8d1cb-5508-45f7-90a8-fe9df8a4dac2%2F7w9cf3b_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the domain of the function. (Enter your answer using interval notation.)**
\[ f(u) = \frac{u + 2}{2 + \frac{2}{u + 2}} \]
---
### Explanation
To find the domain of the function, we need to determine for which values of \( u \) the function is defined. The function is a fraction, and it is undefined when the denominator equals zero.
1. **Denominator Analysis:**
\[
2 + \frac{2}{u + 2} \neq 0
\]
- First, identify where \( \frac{2}{u + 2} \) is undefined:
\[
u + 2 \neq 0 \quad \Rightarrow \quad u \neq -2
\]
- Next, solve for when the entire denominator is zero:
\[
2 + \frac{2}{u + 2} = 0
\]
\[
\frac{2}{u + 2} = -2
\]
\[
2 = -2(u + 2)
\]
\[
2 = -2u - 4
\]
\[
6 = -2u
\]
\[
u = -3
\]
2. **Domain:**
- Therefore, the domain is all real numbers except \( u = -2 \) and \( u = -3 \).
### Domain in Interval Notation
\[
(-\infty, -3) \cup (-3, -2) \cup (-2, \infty)
\]
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