2. Let f(x) be the following piecewise-defined function -x3 – 6, x< -2 { |r|, 6+ x², f (x) = -2 < x < 2 x > 2 (a) Sketch the graph of f(x). Show your work!
2. Let f(x) be the following piecewise-defined function -x3 – 6, x< -2 { |r|, 6+ x², f (x) = -2 < x < 2 x > 2 (a) Sketch the graph of f(x). Show your work!
Algebra and Trigonometry (6th Edition)
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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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![### Piecewise-Defined Function Analysis
#### Problem Statement
Consider the following piecewise-defined function:
\[ f(x) = \begin{cases}
-x^3 - 6, & x < -2 \\
|x|, & -2 \le x \le 2 \\
6 + x^2, & x > 2
\end{cases} \]
Answer the following questions:
**(a) Sketch the graph of \( f(x) \). Show your work!**
**(b) What is the range of \( f(x) \)? Does the graph of \( f(x) \) respect any symmetries?**
**(c) Is there any discontinuity in the graph of \( f(x) \) (particularly at \( x = -2 \) and \( x = 2 \))?**
### Detailed Analysis
#### (a) Sketching the Graph
1. **For \( x < -2 \)**:
- The function is \( f(x) = -x^3 - 6 \).
- This is a cubic function, which will generate a curve that decreases monotonically as \( x \) becomes more negative.
2. **For \( -2 \le x \le 2 \)**:
- The function is \( f(x) = |x| \).
- This is an absolute value function, which creates a V-shaped graph with the vertex at the origin (0, 0).
3. **For \( x > 2 \)**:
- The function is \( f(x) = 6 + x^2 \).
- This is a quadratic function, which generates an upward-opening parabola starting from \( x = 2 \).
To sketch, plot the following:
- The cubic curve for \( x < -2 \).
- The absolute value line for \( -2 \le x \le 2 \).
- The quadratic curve for \( x > 2 \).
#### (b) Range and Symmetry
- **Range**: Identify the minimum and maximum values of the function as \( x \) varies:
- For \( x < -2 \), the function \( f(x) = -x^3 - 6 \) can take very large negative values.
- For \( -2 \le x \le 2 \), the values range from 0 to 2.
- For \( x >](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3ea6570-68e1-449f-84f9-2e2550697ea4%2Fdb735e03-96e4-48a8-a03a-7303d44532d1%2Fpj79fqg.jpeg&w=3840&q=75)
Transcribed Image Text:### Piecewise-Defined Function Analysis
#### Problem Statement
Consider the following piecewise-defined function:
\[ f(x) = \begin{cases}
-x^3 - 6, & x < -2 \\
|x|, & -2 \le x \le 2 \\
6 + x^2, & x > 2
\end{cases} \]
Answer the following questions:
**(a) Sketch the graph of \( f(x) \). Show your work!**
**(b) What is the range of \( f(x) \)? Does the graph of \( f(x) \) respect any symmetries?**
**(c) Is there any discontinuity in the graph of \( f(x) \) (particularly at \( x = -2 \) and \( x = 2 \))?**
### Detailed Analysis
#### (a) Sketching the Graph
1. **For \( x < -2 \)**:
- The function is \( f(x) = -x^3 - 6 \).
- This is a cubic function, which will generate a curve that decreases monotonically as \( x \) becomes more negative.
2. **For \( -2 \le x \le 2 \)**:
- The function is \( f(x) = |x| \).
- This is an absolute value function, which creates a V-shaped graph with the vertex at the origin (0, 0).
3. **For \( x > 2 \)**:
- The function is \( f(x) = 6 + x^2 \).
- This is a quadratic function, which generates an upward-opening parabola starting from \( x = 2 \).
To sketch, plot the following:
- The cubic curve for \( x < -2 \).
- The absolute value line for \( -2 \le x \le 2 \).
- The quadratic curve for \( x > 2 \).
#### (b) Range and Symmetry
- **Range**: Identify the minimum and maximum values of the function as \( x \) varies:
- For \( x < -2 \), the function \( f(x) = -x^3 - 6 \) can take very large negative values.
- For \( -2 \le x \le 2 \), the values range from 0 to 2.
- For \( x >
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