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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where is f discontinuous? explain why
![### Piecewise Function Definition Example
Consider the function \( f(t) \):
\[ f(t) = \begin{cases}
\sqrt{-x} & \text{if } x < 0 \\
3 - x & \text{if } 0 \leq x < 3 \\
(x - 3)^2 & \text{if } x > 3
\end{cases} \]
This piecewise function is defined by three different expressions, each applicable over a certain range of the variable \( x \):
1. **\( \sqrt{-x} \) if \( x < 0 \)**:
- For values of \( x \) that are less than 0, the function is defined by the square root of the negation of \( x \).
2. **\( 3 - x \) if \( 0 \leq x < 3 \)**:
- For values of \( x \) ranging from 0 (inclusive) to less than 3 (exclusive), the function is defined by the linear expression \( 3 - x \).
3. **\( (x - 3)^2 \) if \( x > 3 \)**:
- For values of \( x \) that are greater than 3, the function is defined by the square of the difference between \( x \) and 3.
This function provides an example of how a single function can have different behaviors or rules for different intervals of the domain.
### Visual Representation
To graph this piecewise function:
- For \( x < 0 \), plot the graph of \( \sqrt{-x} \).
- For \( 0 \leq x < 3 \), plot the graph of the linear function \( 3 - x \).
- For \( x > 3 \), plot the graph of the quadratic function \( (x - 3)^2 \).
Each segment of the graph will change at the boundaries \( x = 0 \) and \( x = 3 \), which are the points where the function switches from one expression to another.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76ff96d7-fc10-47d4-9657-45b391b369b2%2F5eb93826-a2cf-4d08-b799-fa79d1a8fc3a%2Fza6qgvy_processed.png&w=3840&q=75)
Transcribed Image Text:### Piecewise Function Definition Example
Consider the function \( f(t) \):
\[ f(t) = \begin{cases}
\sqrt{-x} & \text{if } x < 0 \\
3 - x & \text{if } 0 \leq x < 3 \\
(x - 3)^2 & \text{if } x > 3
\end{cases} \]
This piecewise function is defined by three different expressions, each applicable over a certain range of the variable \( x \):
1. **\( \sqrt{-x} \) if \( x < 0 \)**:
- For values of \( x \) that are less than 0, the function is defined by the square root of the negation of \( x \).
2. **\( 3 - x \) if \( 0 \leq x < 3 \)**:
- For values of \( x \) ranging from 0 (inclusive) to less than 3 (exclusive), the function is defined by the linear expression \( 3 - x \).
3. **\( (x - 3)^2 \) if \( x > 3 \)**:
- For values of \( x \) that are greater than 3, the function is defined by the square of the difference between \( x \) and 3.
This function provides an example of how a single function can have different behaviors or rules for different intervals of the domain.
### Visual Representation
To graph this piecewise function:
- For \( x < 0 \), plot the graph of \( \sqrt{-x} \).
- For \( 0 \leq x < 3 \), plot the graph of the linear function \( 3 - x \).
- For \( x > 3 \), plot the graph of the quadratic function \( (x - 3)^2 \).
Each segment of the graph will change at the boundaries \( x = 0 \) and \( x = 3 \), which are the points where the function switches from one expression to another.
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