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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Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Question
![### Graphing a Piecewise-Defined Function
**Objective:** Learn how to graph a piecewise-defined function.
A piecewise-defined function is expressed with different formulas in different parts of its domain.
Given the piecewise-defined function:
\[ f(x) = \begin{cases}
-3 - x & \text{if } x \leq 2 \\
-6 + 2x & \text{if } x > 2
\end{cases} \]
We are asked to choose the correct graph that represents this function from the four options shown.
#### Graph Options:
**Option A:**
- This graph starts with the line segment \(y = -3 - x\) for \(x \leq 2\) and shows a closed circle at \(x = 2\), \((2, -5)\).
- For \(x > 2\), the graph shows the line segment \(y = -6 + 2x\) and shows an open circle at \(x = 2\), \((2, -2)\).
**Option B:**
- Similar to Option A, this starts with the line segment \(y = -3 - x\) for \(x \leq 2\) and shows a closed circle at \(x = 2\), \((2, -5)\).
- For \(x > 2\), the graph correctly shows \(y = -6 + 2x\), but this time the segment appears with an open circle at \(x = 2\), \((2, -2)\), though overall the segments are mismatched.
**Option C:**
- This graph shows the line segment for \(y = -3 - x\) for \(x \leq 2\), with a closed circle at \(x = 2\), \((2, -5)\), and another incorrect segment following for \(x > 2\).
**Option D:**
- This graph combines the line segment \(y = -3 - x\) for \(x \leq 2\) and the second piece \(y = -6 + 2x\) correctly, but the points and lines are arranged incorrectly.
#### Explanation:
To correctly represent the piecewise function:
1. For \(x \leq 2\), use the equation \(y = -3 - x\).
- At \(x =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89ef0769-2756-4878-bbe7-8a87b2840a03%2Fc75df9fc-f84a-4ad3-9dc3-717a43bf7d7f%2Fllv6s4p_processed.png&w=3840&q=75)
Transcribed Image Text:### Graphing a Piecewise-Defined Function
**Objective:** Learn how to graph a piecewise-defined function.
A piecewise-defined function is expressed with different formulas in different parts of its domain.
Given the piecewise-defined function:
\[ f(x) = \begin{cases}
-3 - x & \text{if } x \leq 2 \\
-6 + 2x & \text{if } x > 2
\end{cases} \]
We are asked to choose the correct graph that represents this function from the four options shown.
#### Graph Options:
**Option A:**
- This graph starts with the line segment \(y = -3 - x\) for \(x \leq 2\) and shows a closed circle at \(x = 2\), \((2, -5)\).
- For \(x > 2\), the graph shows the line segment \(y = -6 + 2x\) and shows an open circle at \(x = 2\), \((2, -2)\).
**Option B:**
- Similar to Option A, this starts with the line segment \(y = -3 - x\) for \(x \leq 2\) and shows a closed circle at \(x = 2\), \((2, -5)\).
- For \(x > 2\), the graph correctly shows \(y = -6 + 2x\), but this time the segment appears with an open circle at \(x = 2\), \((2, -2)\), though overall the segments are mismatched.
**Option C:**
- This graph shows the line segment for \(y = -3 - x\) for \(x \leq 2\), with a closed circle at \(x = 2\), \((2, -5)\), and another incorrect segment following for \(x > 2\).
**Option D:**
- This graph combines the line segment \(y = -3 - x\) for \(x \leq 2\) and the second piece \(y = -6 + 2x\) correctly, but the points and lines are arranged incorrectly.
#### Explanation:
To correctly represent the piecewise function:
1. For \(x \leq 2\), use the equation \(y = -3 - x\).
- At \(x =
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