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,...
Related questions
Question
![Evaluate the function \( f \) at the given value.
\[
f(x) =
\begin{cases}
5x + 1, & \text{if } x < 3 \\
3x, & \text{if } 3 \leq x \leq 5 \\
3 - 5x, & \text{if } x > 5
\end{cases}
\]
\( f(-3) \) for \( f(x) \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc28712d6-8cc5-484b-ae89-313fccbcf9cb%2F8cb27270-a1f9-48fc-b9a7-5165c8ee697b%2Fftn0mmo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Evaluate the function \( f \) at the given value.
\[
f(x) =
\begin{cases}
5x + 1, & \text{if } x < 3 \\
3x, & \text{if } 3 \leq x \leq 5 \\
3 - 5x, & \text{if } x > 5
\end{cases}
\]
\( f(-3) \) for \( f(x) \)
![**Title: Function Representation and Graph Analysis**
**Question: Which graph best represents the function?**
**Function Definition:**
The function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
-4, & \text{if } x \geq 1 \\
-1 - x, & \text{if } x < 1
\end{cases}
\]
**Graph Explanation:**
The graph below features a standard Cartesian coordinate plane:
- **Axes:**
- The horizontal axis is labeled as \( x \).
- The vertical axis is labeled as \( y \).
- **Grid:**
- The grid is marked with small dots to indicate scale and provide guidance for plotting points.
**Interpreting the Function:**
1. **For \( x \geq 1 \):**
- The function \( f(x) = -4 \) is constant. This means for any \( x \) value that is 1 or greater, \( y \) remains at -4.
- On the graph, this part appears as a horizontal line at \( y = -4 \) starting from \( x = 1 \) and extending to the right.
2. **For \( x < 1 \):**
- The function \( f(x) = -1 - x \) is a linear equation with a negative slope.
- This creates a downward sloping line starting from the left towards \( x = 1 \).
- The line’s slope can be calculated as -1, meaning it decreases by 1 unit vertically for each increase of 1 unit horizontally.
**Plotting Specific Points:**
- **Line for \( x < 1 \):**
- If \( x = 0 \), then \( f(0) = -1 \). Plot the point (0, -1).
- If \( x = -1 \), then \( f(-1) = 0 \). Plot the point (-1, 0).
- These points form part of the line for \( x < 1 \) which will intersect the y-axis at \( -1 \).
- **Horizontal Line \( x \geq 1 \):**
- Starts at the point (1, -4).
Understanding this piecewise function helps in discerning its graphical representation, crucial for problem-solving in mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc28712d6-8cc5-484b-ae89-313fccbcf9cb%2F8cb27270-a1f9-48fc-b9a7-5165c8ee697b%2Ft57wgn7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Function Representation and Graph Analysis**
**Question: Which graph best represents the function?**
**Function Definition:**
The function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
-4, & \text{if } x \geq 1 \\
-1 - x, & \text{if } x < 1
\end{cases}
\]
**Graph Explanation:**
The graph below features a standard Cartesian coordinate plane:
- **Axes:**
- The horizontal axis is labeled as \( x \).
- The vertical axis is labeled as \( y \).
- **Grid:**
- The grid is marked with small dots to indicate scale and provide guidance for plotting points.
**Interpreting the Function:**
1. **For \( x \geq 1 \):**
- The function \( f(x) = -4 \) is constant. This means for any \( x \) value that is 1 or greater, \( y \) remains at -4.
- On the graph, this part appears as a horizontal line at \( y = -4 \) starting from \( x = 1 \) and extending to the right.
2. **For \( x < 1 \):**
- The function \( f(x) = -1 - x \) is a linear equation with a negative slope.
- This creates a downward sloping line starting from the left towards \( x = 1 \).
- The line’s slope can be calculated as -1, meaning it decreases by 1 unit vertically for each increase of 1 unit horizontally.
**Plotting Specific Points:**
- **Line for \( x < 1 \):**
- If \( x = 0 \), then \( f(0) = -1 \). Plot the point (0, -1).
- If \( x = -1 \), then \( f(-1) = 0 \). Plot the point (-1, 0).
- These points form part of the line for \( x < 1 \) which will intersect the y-axis at \( -1 \).
- **Horizontal Line \( x \geq 1 \):**
- Starts at the point (1, -4).
Understanding this piecewise function helps in discerning its graphical representation, crucial for problem-solving in mathematics.
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