Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Piecewise Functions: Question #3**
Consider the piecewise function \( f(x) \):
\[
f(x) =
\begin{cases}
x^2 & \text{if } x \leq 0 \\
1 - x & \text{if } x > 0
\end{cases}
\]
**Questions:**
a) **Sketch a graph of the piecewise function**
To graph the piecewise function, plot each piece of the function separately according to the specified intervals:
- For \( x \leq 0 \), plot \( f(x) = x^2 \). This is the graph of a parabola opening upwards on the left side of the y-axis.
- For \( x > 0 \), plot \( f(x) = 1 - x \). This is the graph of a straight line with a slope of -1 that starts from the point where \( x \) is slightly greater than 0.
Make sure to mark any points of connection or discontinuity accurately.
b) **Write the domain in interval notation**
- The domain of \( f(x) \) consists of all the x-values for which the function is defined. According to the definition, \( f(x) \) is defined for all real numbers \( x \).
**Hence, the domain in interval notation is:**
\[ (-\infty, \infty) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ed96b38-bb6c-4f0b-a794-43dac03e0984%2F6531c788-e011-4e2a-8d47-967b28fcfd3b%2Fm9x4k4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Piecewise Functions: Question #3**
Consider the piecewise function \( f(x) \):
\[
f(x) =
\begin{cases}
x^2 & \text{if } x \leq 0 \\
1 - x & \text{if } x > 0
\end{cases}
\]
**Questions:**
a) **Sketch a graph of the piecewise function**
To graph the piecewise function, plot each piece of the function separately according to the specified intervals:
- For \( x \leq 0 \), plot \( f(x) = x^2 \). This is the graph of a parabola opening upwards on the left side of the y-axis.
- For \( x > 0 \), plot \( f(x) = 1 - x \). This is the graph of a straight line with a slope of -1 that starts from the point where \( x \) is slightly greater than 0.
Make sure to mark any points of connection or discontinuity accurately.
b) **Write the domain in interval notation**
- The domain of \( f(x) \) consists of all the x-values for which the function is defined. According to the definition, \( f(x) \) is defined for all real numbers \( x \).
**Hence, the domain in interval notation is:**
\[ (-\infty, \infty) \]
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