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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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
![The image provides instructions to sketch a graph of the piecewise function \( f(x) \), defined as follows:
\[
f(x) = \begin{cases}
-3 & \text{if } x \leq -1 \\
2x - 1 & \text{if } -1 < x \leq 2 \\
2 & \text{if } x > 2
\end{cases}
\]
Additionally, the image includes a coordinate plane (graph grid) with the x-axis and y-axis both labeled from -5 to 5. The grid appears blank with no features or plots.
Here is a detailed explanation of how to sketch the graph of the function \( f(x) \):
1. For \( x \leq -1 \):
- The function \( f(x) = -3 \) is a horizontal line since the output value is always -3 regardless of the input.
- Plot a horizontal line at \( y = -3 \) for all \( x \leq -1 \).
2. For \( -1 < x \leq 2 \):
- The function \( f(x) = 2x - 1 \) is a linear function with a slope of 2 and a y-intercept of -1.
- For \( x = -1 \): \( f(-1) = 2(-1) - 1 = -2 - 1 = -3 \).
- Since this point overlaps with the value at \( x = -1 \) from the previous piece, it should be a continuation.
- For \( x = 2 \): \( f(2) = 2(2) - 1 = 4 - 1 = 3 \). Mark an open circle at \( (2, 3) \) because this value is exclusive for this piece.
- Plot a dashed (indicating it is cut off at the boundaries) line connecting these calculated points (\( (-1, -3) \) to \( (2, 3) \)).
3. For \( x > 2 \):
- The function \( f(x) = 2 \) is constant.
- Plot a horizontal line at \( y = 2 \) for all \( x > 2 \). Start with an open circle at \( (2, 3) \) to show continuity](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48cff491-b173-4746-bb53-6890caeb4c28%2F471f1901-289f-4226-88ba-92938836cd9a%2Fwnjyttd.png&w=3840&q=75)
Transcribed Image Text:The image provides instructions to sketch a graph of the piecewise function \( f(x) \), defined as follows:
\[
f(x) = \begin{cases}
-3 & \text{if } x \leq -1 \\
2x - 1 & \text{if } -1 < x \leq 2 \\
2 & \text{if } x > 2
\end{cases}
\]
Additionally, the image includes a coordinate plane (graph grid) with the x-axis and y-axis both labeled from -5 to 5. The grid appears blank with no features or plots.
Here is a detailed explanation of how to sketch the graph of the function \( f(x) \):
1. For \( x \leq -1 \):
- The function \( f(x) = -3 \) is a horizontal line since the output value is always -3 regardless of the input.
- Plot a horizontal line at \( y = -3 \) for all \( x \leq -1 \).
2. For \( -1 < x \leq 2 \):
- The function \( f(x) = 2x - 1 \) is a linear function with a slope of 2 and a y-intercept of -1.
- For \( x = -1 \): \( f(-1) = 2(-1) - 1 = -2 - 1 = -3 \).
- Since this point overlaps with the value at \( x = -1 \) from the previous piece, it should be a continuation.
- For \( x = 2 \): \( f(2) = 2(2) - 1 = 4 - 1 = 3 \). Mark an open circle at \( (2, 3) \) because this value is exclusive for this piece.
- Plot a dashed (indicating it is cut off at the boundaries) line connecting these calculated points (\( (-1, -3) \) to \( (2, 3) \)).
3. For \( x > 2 \):
- The function \( f(x) = 2 \) is constant.
- Plot a horizontal line at \( y = 2 \) for all \( x > 2 \). Start with an open circle at \( (2, 3) \) to show continuity
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