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
FOR WHAT VALUES OF C IS THIS FUNCTION CONTINUOUS?
![The given function \( f(x) \) is a piecewise function, defined as follows:
\[ f(x) = \begin{cases}
cx^2 & \text{if } x < 1 \\
\frac{6}{c - x} & \text{if } x = 1 \\
6 - cx & \text{if } x > 1
\end{cases} \]
This means that the function \( f(x) \) behaves differently depending on the value of \( x \):
- For values of \( x \) that are less than \( 1 \), \( f(x) \) is given by \( cx^2 \).
- For \( x \) exactly equal to \( 1 \), \( f(x) \) takes the value \( \frac{6}{c - x} \).
- For values of \( x \) that are greater than \( 1 \), \( f(x) \) is given by \( 6 - cx \).
This form of piecewise function is useful in various mathematical and engineering applications where a single equation is not sufficient to describe a system over different ranges of input values.
For an educational website, explaining piecewise functions helps learners understand how complex systems can be broken down into simpler segments that each have their own specific behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F36f9046e-a34d-43f2-ba4c-c4d16262d63e%2F4077cf31-7b21-45b4-9563-3eb6dfae6acc%2F1s2s2o_processed.png&w=3840&q=75)
Transcribed Image Text:The given function \( f(x) \) is a piecewise function, defined as follows:
\[ f(x) = \begin{cases}
cx^2 & \text{if } x < 1 \\
\frac{6}{c - x} & \text{if } x = 1 \\
6 - cx & \text{if } x > 1
\end{cases} \]
This means that the function \( f(x) \) behaves differently depending on the value of \( x \):
- For values of \( x \) that are less than \( 1 \), \( f(x) \) is given by \( cx^2 \).
- For \( x \) exactly equal to \( 1 \), \( f(x) \) takes the value \( \frac{6}{c - x} \).
- For values of \( x \) that are greater than \( 1 \), \( f(x) \) is given by \( 6 - cx \).
This form of piecewise function is useful in various mathematical and engineering applications where a single equation is not sufficient to describe a system over different ranges of input values.
For an educational website, explaining piecewise functions helps learners understand how complex systems can be broken down into simpler segments that each have their own specific behavior.
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