the instructions may just say, "Graph. Show all pertinent information such as intercepts, asymptotes local extema, inflection point", ilections Domain: x-intercept(s): y-intercept(s):_ (1). For the function f(x)= x² X Find each of the following (if they exist). Show the work yielding your answers: Horizontal Asymptote. Inflection Points -7 -6 -5 -4 Vertical Asymptotes, and limit approaching the vertical asymptote from each side. -3 -2 -8 Points on the graph corresponding to local extrema (max or min?) 7 -6 -5 -4 4 -3 2 1 -10 --1 --2 --3- --4 --5- B 2 - 1 +3 2 :.. **** 3 4 5 6 In 7 8 9 10 11 12 13
the instructions may just say, "Graph. Show all pertinent information such as intercepts, asymptotes local extema, inflection point", ilections Domain: x-intercept(s): y-intercept(s):_ (1). For the function f(x)= x² X Find each of the following (if they exist). Show the work yielding your answers: Horizontal Asymptote. Inflection Points -7 -6 -5 -4 Vertical Asymptotes, and limit approaching the vertical asymptote from each side. -3 -2 -8 Points on the graph corresponding to local extrema (max or min?) 7 -6 -5 -4 4 -3 2 1 -10 --1 --2 --3- --4 --5- B 2 - 1 +3 2 :.. **** 3 4 5 6 In 7 8 9 10 11 12 13
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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![---
**Graphing Rational Functions**
In this exercise, we are provided with a function \( f(x) = \frac{4}{x^2} - \frac{2}{x} + 3 \). We are instructed to find and graph several key features of this function, including domain, intercepts, asymptotes, extrema, and inflection points.
### Function Analysis
Given function: \( f(x) = \frac{4}{x^2} - \frac{2}{x} + 3 \)
1. **Domain**:
\[
\text{Domain: } \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\]
The domain is the set of all possible input values \( x \) that do not cause the function to be undefined.
2. **x-intercept(s)**:
\[
\text{x-intercept(s): } \_\_\_\_\_\_\_\_\_\_\_\_\_
\]
These are the points where the function \( f(x) \) crosses the x-axis, where \( f(x) = 0 \).
3. **y-intercept(s)**:
\[
\text{y-intercept(s): } \_\_\_\_\_\_\_\_\_\_\_\_
\]
These are the points where the function \( f(x) \) crosses the y-axis, where \( x = 0 \).
4. **Vertical Asymptotes** and the **limit** approaching the vertical asymptote from each side:
\[
\text{Vertical Asymptotes: } \_\_\_\_\_\_\_\_\_\_\_\_\_
\]
Vertical asymptotes occur where the denominator is zero and the function value approaches infinity.
5. **Horizontal Asymptote**:
\[
\text{Horizontal Asymptote: } \_\_\_\_\_\_\_\_
\]
The horizontal asymptote is the value that the function approaches as \( x \) approaches infinity.
6. Points on the graph corresponding to local **extrema** (max or min?):
\[
\text{Local Extrema: } \_\_\_\_\_\_\_\_\_\_\_
\]
Local extrema are the local maximum or minimum points on the function.
7](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99284a24-5b6d-4c49-8bfd-674d8761bb54%2F1e91b67c-e8ac-4722-9eca-a264beb49ab7%2Fr6t3ajl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Graphing Rational Functions**
In this exercise, we are provided with a function \( f(x) = \frac{4}{x^2} - \frac{2}{x} + 3 \). We are instructed to find and graph several key features of this function, including domain, intercepts, asymptotes, extrema, and inflection points.
### Function Analysis
Given function: \( f(x) = \frac{4}{x^2} - \frac{2}{x} + 3 \)
1. **Domain**:
\[
\text{Domain: } \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\]
The domain is the set of all possible input values \( x \) that do not cause the function to be undefined.
2. **x-intercept(s)**:
\[
\text{x-intercept(s): } \_\_\_\_\_\_\_\_\_\_\_\_\_
\]
These are the points where the function \( f(x) \) crosses the x-axis, where \( f(x) = 0 \).
3. **y-intercept(s)**:
\[
\text{y-intercept(s): } \_\_\_\_\_\_\_\_\_\_\_\_
\]
These are the points where the function \( f(x) \) crosses the y-axis, where \( x = 0 \).
4. **Vertical Asymptotes** and the **limit** approaching the vertical asymptote from each side:
\[
\text{Vertical Asymptotes: } \_\_\_\_\_\_\_\_\_\_\_\_\_
\]
Vertical asymptotes occur where the denominator is zero and the function value approaches infinity.
5. **Horizontal Asymptote**:
\[
\text{Horizontal Asymptote: } \_\_\_\_\_\_\_\_
\]
The horizontal asymptote is the value that the function approaches as \( x \) approaches infinity.
6. Points on the graph corresponding to local **extrema** (max or min?):
\[
\text{Local Extrema: } \_\_\_\_\_\_\_\_\_\_\_
\]
Local extrema are the local maximum or minimum points on the function.
7
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