For each graph of f shown below, answer parts (a)-(d). -3 domain range -2 -1 y 15 10 5 1 f(x) = x¹ (a) Find the domain and range of f. (Enter your answer using interval notation.) O 2 3 Et X (b) Find the x- and y-intercepts of the graph of f. (If an answer does not exist, enter DNE.) x-intercept (x, y) = y-intercept (x, y) = ( (c) Determine the open intervals on which f is increasing, decreasing, or constant. (Enter your answers using inter enter DNE.) Type here to search
For each graph of f shown below, answer parts (a)-(d). -3 domain range -2 -1 y 15 10 5 1 f(x) = x¹ (a) Find the domain and range of f. (Enter your answer using interval notation.) O 2 3 Et X (b) Find the x- and y-intercepts of the graph of f. (If an answer does not exist, enter DNE.) x-intercept (x, y) = y-intercept (x, y) = ( (c) Determine the open intervals on which f is increasing, decreasing, or constant. (Enter your answers using inter enter DNE.) Type here to search
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2.4 14
![### Analyzing Function Behavior and Symmetry
#### Questions:
**(c) Determine the open intervals on which \( f \) is increasing, decreasing, or constant.**
*(Enter your answers using interval notation. If an answer does not exist, enter DNE.)*
- **Increasing**:
- [Input interval here]
- **Decreasing**:
- [Input interval here]
- **Constant**:
- [Input interval here]
**(d) Determine whether \( f \) is even, odd, or neither.**
- \( \bigcirc \) even
- \( \bigcirc \) \[X\] odd
- \( \bigcirc \) neither
**Describe the symmetry:**
\[ \boxed{\text{x}} \] x-axis symmetry
\[ \boxed{} \] y-axis symmetry
\[ \boxed{} \] origin symmetry
\[ \boxed{} \] symmetry over the line \( y = x \)
\[ \boxed{} \] no symmetry
#### Explanation of Concepts:
**1. Determining Intervals:**
- **Increasing Interval:** A function \( f(x) \) is increasing on an interval if, for every \( x_1 \) and \( x_2 \) in the interval, \( x_1 < x_2 \) implies \( f(x_1) < f(x_2) \).
- **Decreasing Interval:** A function \( f(x) \) is decreasing on an interval if, for every \( x_1 \) and \( x_2 \) in the interval, \( x_1 < x_2 \) implies \( f(x_1) > f(x_2) \).
- **Constant Interval:** A function \( f(x) \) is constant on an interval if, for every \( x_1 \) and \( x_2 \) in the interval, \( f(x_1) = f(x_2) \).
**2. Function Symmetry:**
- **Even Function:** A function \( f(x) \) is even if \( f(x) = f(-x) \) for all \( x \) in the domain. It is symmetric about the y-axis.
- **Odd Function:** A function \( f(x) \) is odd if \( -f(x) = f(-x) \) for all \( x \) in the domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86020f22-4efe-437b-8b05-308770976ff7%2Ffb69b0a5-4704-4644-9095-87f30bf0ece8%2Fkhkjujq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Analyzing Function Behavior and Symmetry
#### Questions:
**(c) Determine the open intervals on which \( f \) is increasing, decreasing, or constant.**
*(Enter your answers using interval notation. If an answer does not exist, enter DNE.)*
- **Increasing**:
- [Input interval here]
- **Decreasing**:
- [Input interval here]
- **Constant**:
- [Input interval here]
**(d) Determine whether \( f \) is even, odd, or neither.**
- \( \bigcirc \) even
- \( \bigcirc \) \[X\] odd
- \( \bigcirc \) neither
**Describe the symmetry:**
\[ \boxed{\text{x}} \] x-axis symmetry
\[ \boxed{} \] y-axis symmetry
\[ \boxed{} \] origin symmetry
\[ \boxed{} \] symmetry over the line \( y = x \)
\[ \boxed{} \] no symmetry
#### Explanation of Concepts:
**1. Determining Intervals:**
- **Increasing Interval:** A function \( f(x) \) is increasing on an interval if, for every \( x_1 \) and \( x_2 \) in the interval, \( x_1 < x_2 \) implies \( f(x_1) < f(x_2) \).
- **Decreasing Interval:** A function \( f(x) \) is decreasing on an interval if, for every \( x_1 \) and \( x_2 \) in the interval, \( x_1 < x_2 \) implies \( f(x_1) > f(x_2) \).
- **Constant Interval:** A function \( f(x) \) is constant on an interval if, for every \( x_1 \) and \( x_2 \) in the interval, \( f(x_1) = f(x_2) \).
**2. Function Symmetry:**
- **Even Function:** A function \( f(x) \) is even if \( f(x) = f(-x) \) for all \( x \) in the domain. It is symmetric about the y-axis.
- **Odd Function:** A function \( f(x) \) is odd if \( -f(x) = f(-x) \) for all \( x \) in the domain.
![**For each graph of function \( f \) shown below, answer parts (a)-(d).**
**Graph Description:**
- The graph represents the function \( f(x) = x^4 \).
- The \( x \)-axis ranges from \(-3\) to \(3\).
- The \( y \)-axis ranges from \(-5\) to \(20\).
- The graph is symmetric with respect to the \( y \)-axis and features a paraboloid shape opening upwards, with the vertex at the origin \((0,0)\).
- Specific points marked along the \( y \)-axis include \(5\), \(10\), and \(15\).
**Question (a): Find the domain and range of \( f \). (Enter your answer using interval notation.)**
- **Domain:** \([ \, \_\_\_\_ , \_\_\_\_ \, ]\)
- **Range:** \([ \, \_\_\_\_ , \_\_\_\_ \, ]\)
**Question (b): Find the \( x \)- and \( y \)-intercepts of the graph of \( f \). (If an answer does not exist, enter DNE.)**
- **\( x \)-intercept:** \((x, y) = (\_\_\_\_, \_\_\_\_) \)
- **\( y \)-intercept:** \((x, y) = (\_\_\_\_, \_\_\_\_) \)
**Question (c): Determine the open intervals on which \( f \) is increasing, decreasing, or constant. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)**
- **Increasing:** \(\_\_\_\_\)
- **Decreasing:** \(\_\_\_\_\)
- **Constant:** \(\_\_\_\_\)
*(Answers:)*
**(a)**
- **Domain:** \((-\infty, \infty)\)
- **Range:** \([0, \infty)\)
**(b)**
- **\( x \)-intercept:** \((0,0)\)
- **\( y \)-intercept:** \((0,0)\)
**(c)**
- **Increasing:** \((0, \infty)\)
- **Decreasing:** \((-\infty, 0)\)
- **Constant:** DNE](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86020f22-4efe-437b-8b05-308770976ff7%2Ffb69b0a5-4704-4644-9095-87f30bf0ece8%2Fpu9eycf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**For each graph of function \( f \) shown below, answer parts (a)-(d).**
**Graph Description:**
- The graph represents the function \( f(x) = x^4 \).
- The \( x \)-axis ranges from \(-3\) to \(3\).
- The \( y \)-axis ranges from \(-5\) to \(20\).
- The graph is symmetric with respect to the \( y \)-axis and features a paraboloid shape opening upwards, with the vertex at the origin \((0,0)\).
- Specific points marked along the \( y \)-axis include \(5\), \(10\), and \(15\).
**Question (a): Find the domain and range of \( f \). (Enter your answer using interval notation.)**
- **Domain:** \([ \, \_\_\_\_ , \_\_\_\_ \, ]\)
- **Range:** \([ \, \_\_\_\_ , \_\_\_\_ \, ]\)
**Question (b): Find the \( x \)- and \( y \)-intercepts of the graph of \( f \). (If an answer does not exist, enter DNE.)**
- **\( x \)-intercept:** \((x, y) = (\_\_\_\_, \_\_\_\_) \)
- **\( y \)-intercept:** \((x, y) = (\_\_\_\_, \_\_\_\_) \)
**Question (c): Determine the open intervals on which \( f \) is increasing, decreasing, or constant. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)**
- **Increasing:** \(\_\_\_\_\)
- **Decreasing:** \(\_\_\_\_\)
- **Constant:** \(\_\_\_\_\)
*(Answers:)*
**(a)**
- **Domain:** \((-\infty, \infty)\)
- **Range:** \([0, \infty)\)
**(b)**
- **\( x \)-intercept:** \((0,0)\)
- **\( y \)-intercept:** \((0,0)\)
**(c)**
- **Increasing:** \((0, \infty)\)
- **Decreasing:** \((-\infty, 0)\)
- **Constant:** DNE
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