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
![**Educational Content on Comparing and Building Functions**
---
### Pretest: Comparing and Building Functions
**Graph Representation of Function \( f(x) \):**
The graph of the function \( f(x) \) is shown on a coordinate plane. The x-axis ranges from -8 to 8, and the y-axis ranges from -2 to 8. The plot shows a blue line piecewise defined with different slopes. Here are the key points and trends:
- The graph starts at the point (-8, 0), rises to the highest point at (0, 4), then continues through (2, 6), reaches its peak at (4, 2), and finally moves downwards as it extends beyond the visible axes.
**Function \( g(x) \) Definition:**
The function \( g(x) \) is defined by a piecewise function as follows:
\[
g(x) =
\begin{cases}
x - 4 & ; \quad x \leq 1 \\
3 & ; \quad 1 < x \leq 3 \\
x - 6 & ; \quad x > 3
\end{cases}
\]
**Question:**
Which of the following statements is true?
A. The y-intercept of \( g(x) \) is greater than the y-intercept of \( f(x) \).
B. The y-intercept of \( f(x) \) is equal to the y-intercept of \( g(x) \).
C. The y-intercept of \( g(x) \) cannot be determined from the given information.
D. The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \).
---
**Explanation of Graph and Functions:**
To determine the correct answer, we need to compare the y-intercepts of \( f(x) \) and \( g(x) \):
- **For \( f(x) \):** The y-intercept is the value of \( f(0) \). According to the graph, at \( x = 0 \), the value of \( f(x) = 4 \). Hence, the y-intercept of \( f(x) \) is 4.
- **For \( g(x) \):** From the piecewise definition,
\[
g(x) = x - 4 \text{ when } x \leq 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18451349-f722-4343-9ced-dbc580baaa92%2F4610eb71-3aea-4983-a3a8-c2823d13ff98%2Fjc5ijs_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content on Comparing and Building Functions**
---
### Pretest: Comparing and Building Functions
**Graph Representation of Function \( f(x) \):**
The graph of the function \( f(x) \) is shown on a coordinate plane. The x-axis ranges from -8 to 8, and the y-axis ranges from -2 to 8. The plot shows a blue line piecewise defined with different slopes. Here are the key points and trends:
- The graph starts at the point (-8, 0), rises to the highest point at (0, 4), then continues through (2, 6), reaches its peak at (4, 2), and finally moves downwards as it extends beyond the visible axes.
**Function \( g(x) \) Definition:**
The function \( g(x) \) is defined by a piecewise function as follows:
\[
g(x) =
\begin{cases}
x - 4 & ; \quad x \leq 1 \\
3 & ; \quad 1 < x \leq 3 \\
x - 6 & ; \quad x > 3
\end{cases}
\]
**Question:**
Which of the following statements is true?
A. The y-intercept of \( g(x) \) is greater than the y-intercept of \( f(x) \).
B. The y-intercept of \( f(x) \) is equal to the y-intercept of \( g(x) \).
C. The y-intercept of \( g(x) \) cannot be determined from the given information.
D. The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \).
---
**Explanation of Graph and Functions:**
To determine the correct answer, we need to compare the y-intercepts of \( f(x) \) and \( g(x) \):
- **For \( f(x) \):** The y-intercept is the value of \( f(0) \). According to the graph, at \( x = 0 \), the value of \( f(x) = 4 \). Hence, the y-intercept of \( f(x) \) is 4.
- **For \( g(x) \):** From the piecewise definition,
\[
g(x) = x - 4 \text{ when } x \leq 1
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