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
**(b)** Let \( f(x) =
\begin{cases}
3^{\frac{3}{x+1}} & , \, x < 1 \\
2x + 3b & , \, x \geq 1
\end{cases}
\).
Find the constant \( b \) such that the function \( f \) is continuous everywhere.
### Explanation:
To ensure that the function \( f \) is continuous everywhere, the following condition must be met:
\[ \lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x) = f(c) \]
For the point where \( x = 1 \):
\[ x = 1 \]
This means that:
\[ \lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^+}} f(x) = f(1) \]
Hence, the values from both piecewise sections at \( x = 1 \) need to be equal.
\( 1. 1 = \text{RL} = \text{f(x)} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2cc04b1e-803c-4d9d-b550-d2ae04bcc2b3%2Ffb1e24e6-a3bf-453b-ad51-f014241f9b88%2Fze9d9jd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**(b)** Let \( f(x) =
\begin{cases}
3^{\frac{3}{x+1}} & , \, x < 1 \\
2x + 3b & , \, x \geq 1
\end{cases}
\).
Find the constant \( b \) such that the function \( f \) is continuous everywhere.
### Explanation:
To ensure that the function \( f \) is continuous everywhere, the following condition must be met:
\[ \lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x) = f(c) \]
For the point where \( x = 1 \):
\[ x = 1 \]
This means that:
\[ \lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^+}} f(x) = f(1) \]
Hence, the values from both piecewise sections at \( x = 1 \) need to be equal.
\( 1. 1 = \text{RL} = \text{f(x)} \)
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