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
![**Function Definition:**
The piecewise function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
x^2 - 4 & \text{for } x \leq 2 \\
2x - 4 & \text{for } x > 2
\end{cases}
\]
**Graph Explanation:**
- **Axes:** The graph is drawn on a Cartesian coordinate plane with a horizontal \( x \)-axis and a vertical \( y \)-axis.
- **Piecewise Conditions:**
- For \( x \leq 2 \), the function follows the parabola \( y = x^2 - 4 \).
- For \( x > 2 \), the function follows the line \( y = 2x - 4 \).
- **Break Point:** The transition occurs at \( x = 2 \), where the definition of the function changes from the quadratic expression to the linear expression.
**Graph Characteristics:**
- **Quadratic Segment (\( x \leq 2 \)):**
- This segment is part of a parabola opening upwards.
- The vertex on this segment would appear at a point less than or equal to \( x = 2 \).
- **Linear Segment (\( x > 2 \)):**
- This is a straight line with a slope of 2, indicating a rise of 2 units for every 1 unit increase in \( x \).
- **Domain and Range:**
- The domain of the function is all real numbers, divided into two intervals: \( x \leq 2 \) and \( x > 2 \).
- The range depends on the values taken by each piece of the function over its respective interval.
This function's graph will help in understanding how piecewise functions operate differently over different domains.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a54d7d8-382f-4dae-bfd8-b87945b99138%2Fe50cac19-1283-4e50-9456-c426f9d466a9%2F8c2z53_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Function Definition:**
The piecewise function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
x^2 - 4 & \text{for } x \leq 2 \\
2x - 4 & \text{for } x > 2
\end{cases}
\]
**Graph Explanation:**
- **Axes:** The graph is drawn on a Cartesian coordinate plane with a horizontal \( x \)-axis and a vertical \( y \)-axis.
- **Piecewise Conditions:**
- For \( x \leq 2 \), the function follows the parabola \( y = x^2 - 4 \).
- For \( x > 2 \), the function follows the line \( y = 2x - 4 \).
- **Break Point:** The transition occurs at \( x = 2 \), where the definition of the function changes from the quadratic expression to the linear expression.
**Graph Characteristics:**
- **Quadratic Segment (\( x \leq 2 \)):**
- This segment is part of a parabola opening upwards.
- The vertex on this segment would appear at a point less than or equal to \( x = 2 \).
- **Linear Segment (\( x > 2 \)):**
- This is a straight line with a slope of 2, indicating a rise of 2 units for every 1 unit increase in \( x \).
- **Domain and Range:**
- The domain of the function is all real numbers, divided into two intervals: \( x \leq 2 \) and \( x > 2 \).
- The range depends on the values taken by each piece of the function over its respective interval.
This function's graph will help in understanding how piecewise functions operate differently over different domains.
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