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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![The task is to sketch a graph of the function \( f(x) \), which is defined as follows:
\[
f(x) =
\begin{cases}
-3 & \text{if } x \leq -2 \\
2x + 1 & \text{if } -2 < x \leq 2 \\
2 & \text{if } x > 2
\end{cases}
\]
### Graph Details:
1. **Section 1**: For \( x \leq -2 \), the function \( f(x) = -3 \) is constant. This appears as a horizontal line on the graph at \( y = -3 \).
2. **Section 2**: For \( -2 < x \leq 2 \), the function is linear: \( f(x) = 2x + 1 \). This line has a slope of 2 and a y-intercept at \( y = 1 \). It starts just to the right of \( x = -2 \) and ends at \( x = 2 \).
3. **Section 3**: For \( x > 2 \), the function \( f(x) = 2 \) is a constant. This appears as another horizontal line on the graph at \( y = 2 \).
### Graph Sketching Details:
- The y-axis ranges from -5 to 5, and the x-axis ranges from -5 to 5.
- Critical points to note are:
- At \( x = -2 \), the graph transitions from a constant value of -3 to the linear section. There is a jump in function value.
- The linear section spans from \( x = -2 \) to \( x = 2 \), crossing through points such as \( (-2, -3) \), \( (0, 1) \), and \( (2, 5) \).
- At \( x = 2 \), the graph transitions from the linear section to a constant value of 2, creating another jump in the graph.
By following these parameters, the graph can be accurately sketched to represent the piecewise function given.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff29755f2-3a6c-417d-b3ad-401ba07baf3b%2Febee6892-b678-44c4-b548-c6c347bb57fd%2Fskodzuk.jpeg&w=3840&q=75)
Transcribed Image Text:The task is to sketch a graph of the function \( f(x) \), which is defined as follows:
\[
f(x) =
\begin{cases}
-3 & \text{if } x \leq -2 \\
2x + 1 & \text{if } -2 < x \leq 2 \\
2 & \text{if } x > 2
\end{cases}
\]
### Graph Details:
1. **Section 1**: For \( x \leq -2 \), the function \( f(x) = -3 \) is constant. This appears as a horizontal line on the graph at \( y = -3 \).
2. **Section 2**: For \( -2 < x \leq 2 \), the function is linear: \( f(x) = 2x + 1 \). This line has a slope of 2 and a y-intercept at \( y = 1 \). It starts just to the right of \( x = -2 \) and ends at \( x = 2 \).
3. **Section 3**: For \( x > 2 \), the function \( f(x) = 2 \) is a constant. This appears as another horizontal line on the graph at \( y = 2 \).
### Graph Sketching Details:
- The y-axis ranges from -5 to 5, and the x-axis ranges from -5 to 5.
- Critical points to note are:
- At \( x = -2 \), the graph transitions from a constant value of -3 to the linear section. There is a jump in function value.
- The linear section spans from \( x = -2 \) to \( x = 2 \), crossing through points such as \( (-2, -3) \), \( (0, 1) \), and \( (2, 5) \).
- At \( x = 2 \), the graph transitions from the linear section to a constant value of 2, creating another jump in the graph.
By following these parameters, the graph can be accurately sketched to represent the piecewise function given.
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