Homework Help is Here – Start Your Trial Now!
Math
Algebra
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 & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 54E
See similar textbooks
icon
Related questions
Question

please show allll work and help asap

**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
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
Expert Solution
steps

Step by step

Solved in 5 steps with 6 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Basics (types, similarity, etc)
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
College Algebra
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS