Graph the ration function, find the domain of the graph, find the intercepts, find any hole, f (x- 1)2 (x² – 1) any asymptotes, and find additional points on the graph. f(x) =
Graph the ration function, find the domain of the graph, find the intercepts, find any hole, f (x- 1)2 (x² – 1) any asymptotes, and find additional points on the graph. f(x) =
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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![**Graphing Rational Functions and Analyzing Their Properties**
**Objective:**
To graph the rational function \( f(x) = \dfrac{(x-1)^2}{(x^2-1)} \), determine its domain, identify its intercepts, locate any holes, find asymptotes, and determine additional points on the graph.
**Instructions:**
1. **Plot the Rational Function:**
- Use the provided grid to plot the function \( f(x) = \dfrac{(x-1)^2}{(x^2-1)} \).
2. **Find the Domain:**
- Identify all x-values for which the function is defined.
3. **Determine the Intercepts:**
- Locate any points where the graph intersects the x-axis and y-axis.
4. **Identify any Holes in the Graph:**
- Establish if there are any points where the function is undefined due to possible cancellation of factors in the numerator and denominator.
5. **Find the Asymptotes:**
- Determine any vertical or horizontal asymptotes.
6. **Find Additional Points:**
- Calculate and plot additional points to help accurately sketch the graph.
**Graph Analysis:**
- The function \( f(x) = \dfrac{(x-1)^2}{(x^2-1)} \) can be rewritten as \( f(x) = \dfrac{(x-1)^2}{(x-1)(x+1)} \).
- Simplify the expression, noting possible points of discontinuity:
\[ f(x) = \dfrac{x-1}{x+1} \quad \text{for} \quad x \neq \pm 1 \]
**Graph Elements:**
1. **Domain:**
- The function is undefined where the denominator is zero, i.e., for \( x = \pm 1 \).
- Therefore, the domain is \( x \in (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).
2. **Intercepts:**
- **Y-intercept:** Set \( x = 0 \):
\[ f(0) = \dfrac{(0-1)^2}{(0^2-1)} = \dfrac{1}{-1} = -1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7921e840-b4be-4997-af5c-a45fa0acee7f%2Fd558f77e-ce3f-47ef-a497-9abeb1ae6f47%2Fy6e97yi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Graphing Rational Functions and Analyzing Their Properties**
**Objective:**
To graph the rational function \( f(x) = \dfrac{(x-1)^2}{(x^2-1)} \), determine its domain, identify its intercepts, locate any holes, find asymptotes, and determine additional points on the graph.
**Instructions:**
1. **Plot the Rational Function:**
- Use the provided grid to plot the function \( f(x) = \dfrac{(x-1)^2}{(x^2-1)} \).
2. **Find the Domain:**
- Identify all x-values for which the function is defined.
3. **Determine the Intercepts:**
- Locate any points where the graph intersects the x-axis and y-axis.
4. **Identify any Holes in the Graph:**
- Establish if there are any points where the function is undefined due to possible cancellation of factors in the numerator and denominator.
5. **Find the Asymptotes:**
- Determine any vertical or horizontal asymptotes.
6. **Find Additional Points:**
- Calculate and plot additional points to help accurately sketch the graph.
**Graph Analysis:**
- The function \( f(x) = \dfrac{(x-1)^2}{(x^2-1)} \) can be rewritten as \( f(x) = \dfrac{(x-1)^2}{(x-1)(x+1)} \).
- Simplify the expression, noting possible points of discontinuity:
\[ f(x) = \dfrac{x-1}{x+1} \quad \text{for} \quad x \neq \pm 1 \]
**Graph Elements:**
1. **Domain:**
- The function is undefined where the denominator is zero, i.e., for \( x = \pm 1 \).
- Therefore, the domain is \( x \in (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).
2. **Intercepts:**
- **Y-intercept:** Set \( x = 0 \):
\[ f(0) = \dfrac{(0-1)^2}{(0^2-1)} = \dfrac{1}{-1} = -1
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