Difference between revisions of "005 Sample Final A, Question 20"

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(Created page with "'''Question ''' Consider the following rational function, <center><math>f(x) = \frac{x^2+x-2}{x^2-1}</math></center> <br>      a. What is the domain of f?...")
 
 
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<center><math>f(x) = \frac{x^2+x-2}{x^2-1}</math></center> <br>
 
<center><math>f(x) = \frac{x^2+x-2}{x^2-1}</math></center> <br>
  
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&nbsp;&nbsp;&nbsp;&nbsp; a. What is the domain of f? <br>
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:: a. What is the domain of f? <br>
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&nbsp;&nbsp;&nbsp;&nbsp; b. What are the x and y-intercepts of f? <br>
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:: b. What are the x and y-intercepts of f? <br>
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&nbsp;&nbsp;&nbsp;&nbsp; c. What are the vertical and horizontal asymptotes of f, if any? Does f have any holes? <br>
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:: c. What are the vertical and horizontal asymptotes of f, if any? Does f have any holes? <br>
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&nbsp;&nbsp;&nbsp;&nbsp; d. Graph f(x). Make sure to include the information you found above.
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:: d. Graph f(x). Make sure to include the information you found above.
  
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answers
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!Foundations: &nbsp;
 
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|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
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|1) What points are not in the domain of f(x)?
 
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|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
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|2) How do you find the intercepts?
 
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|c) False. <math>y = x^2</math> does not have an inverse.
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|3) How do you find the asymptotes and zeros?
 
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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|4) How do you determine if f has any holes?
 
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|e) True.
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|Answer:
 
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|f) False.  
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|1) The point that are not in the domain of f(x) are zeros of the denominator.
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|2) To find the x-intercepts set y = 0 and solve for x. For y-intercepts set x = 0 and simplify.
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|3) For zeros, find the zeros of the numerator. Vertical asymptotes correspond to zeros of the denominator. Horizontal asymptotes correspond to taking the limit as x goes to <math>\infty</math>
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|4) Holes occur when a single value of x is a zero of both the numerator and denominator.
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Solution:
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 1: &nbsp;
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|-
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|We start by finding the zeros of the denominator since this will give us information about vertical asymptotes and the domain. The zeros of the denominator are x = -1, 1. This tells us the domain is <math>(-\infty, -1) \cup (-1, 1) \cup (1, \infty)</math> and the potential vertical asymptotes are x = -1 and x = 1.
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 2: &nbsp;
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|-
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|Now we should find the y-intercepts(zeros) to determine if f has any holes, which are zeros of numerator and denominator. Thus, y-intercepts correspond to zeros of the numerator x = -2, 1. Now we know we have a hole at x = 1, and a y-intercept at <math>(-2, 0).</math>
 
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 3: &nbsp;
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|-
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|For the horizontal asymptote take the limit as x goes to <math>\infty.</math> This tells us that the horizontal asymptote is y = 1.
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Final Answer: &nbsp;
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|Domain: <math>(-\infty, -1)\cup(-1, 1)\cup (1, \infty)</math> The x-intercept is <math>(-2, 0)</math> and y-intercept is <math>(0, 2)</math>
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|f does have a hole, with vertical asymptote at x = -1, and horizontal asymptote y = 1.
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|[[File:5_Sample_Final_20.png]]
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|}
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[[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]
 
[[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 19:18, 2 June 2015

Question Consider the following rational function,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \frac{x^2+x-2}{x^2-1}}


a. What is the domain of f?
b. What are the x and y-intercepts of f?
c. What are the vertical and horizontal asymptotes of f, if any? Does f have any holes?
d. Graph f(x). Make sure to include the information you found above.


Foundations:  
1) What points are not in the domain of f(x)?
2) How do you find the intercepts?
3) How do you find the asymptotes and zeros?
4) How do you determine if f has any holes?
Answer:
1) The point that are not in the domain of f(x) are zeros of the denominator.
2) To find the x-intercepts set y = 0 and solve for x. For y-intercepts set x = 0 and simplify.
3) For zeros, find the zeros of the numerator. Vertical asymptotes correspond to zeros of the denominator. Horizontal asymptotes correspond to taking the limit as x goes to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty}
4) Holes occur when a single value of x is a zero of both the numerator and denominator.

Solution:

Step 1:  
We start by finding the zeros of the denominator since this will give us information about vertical asymptotes and the domain. The zeros of the denominator are x = -1, 1. This tells us the domain is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1) \cup (-1, 1) \cup (1, \infty)} and the potential vertical asymptotes are x = -1 and x = 1.
Step 2:  
Now we should find the y-intercepts(zeros) to determine if f has any holes, which are zeros of numerator and denominator. Thus, y-intercepts correspond to zeros of the numerator x = -2, 1. Now we know we have a hole at x = 1, and a y-intercept at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-2, 0).}
Step 3:  
For the horizontal asymptote take the limit as x goes to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty.} This tells us that the horizontal asymptote is y = 1.
Final Answer:  
Domain: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1)\cup(-1, 1)\cup (1, \infty)} The x-intercept is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-2, 0)} and y-intercept is
f does have a hole, with vertical asymptote at x = -1, and horizontal asymptote y = 1.
5 Sample Final 20.png


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