Questions on Algebra: Functions, Domain, NOT graphing answered by real tutors!

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Question 151671This question is from textbook
: Can someone please help I don't understand. Find the equations for the horizontal and vertical asymptotes of the following.
A. f(x)=2x+1/x-4
B. g(x)=3x/x^2-4
What is the horizontal vertical for both equations? Thank you This question is from textbook
: Can someone please help I don't understand. Find the equations for the horizontal and vertical asymptotes of the following.
A. f(x)=2x+1/x-4
B. g(x)=3x/x^2-4
What is the horizontal vertical for both equations? Thank you
Answer by jim_thompson5910(9166) About Me  (Show Source):
You can put this solution on YOUR website!
A)



y=(2x+1)/(x-4)) Start with the given function



Looking at the numerator 2x+1, we can see that the degree is 1 since the highest exponent of the numerator is 1. For the denominator x-4, we can see that the degree is 1 since the highest exponent of the denominator is 1.


Horizontal Asymptote:
Since the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure:

To find the horizontal asymptote, first we need to find the leading coefficients of the numerator and the denominator.

Looking at the numerator 2x+1, the leading coefficient is 2

Looking at the denominator x-4, the leading coefficient is 1

So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide 2 by 1 to get (2)/(1)=2


So the horizontal asymptote is y=2





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Vertical Asymptote:
To find the vertical asymptote, just set the denominator equal to zero and solve for x

x-4=0 Set the denominator equal to zero


x=0+4Add 4 to both sides


x=4 Combine like terms on the right side


So the vertical asymptote is x=4


Notice if we graph y=(2x+1)/(x-4), we can visually verify our answers:

drawing(500,500,-10,10,-10,10,<BR> graph(500,500,-10,10,-10,10,(2x+1)/(x-4)),<BR> blue(line(-20,2,20,2)),<BR> green(line(4,-20,4,20))<BR> ) Graph of y=(2x+1)/(x-4)) with the horizontal asymptote y=2 (blue line) and the vertical asymptote x=4 (green line)





B)



y=(3x)/(x^2-4)) Start with the given function



Looking at the numerator 3x, we can see that the degree is 1 since the highest exponent of the numerator is 1. For the denominator x^2-4, we can see that the degree is 2 since the highest exponent of the denominator is 2.


Horizontal Asymptote:

Since the degree of the numerator (which is 1) is less than the degree of the denominator (which is 2), the horizontal asymptote is always y=0

So the horizontal asymptote is y=0



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Vertical Asymptote:
To find the vertical asymptote, just set the denominator equal to zero and solve for x

x^2-4=0 Set the denominator equal to zero


x^2=0+4Add 4 to both sides


x^2=4 Combine like terms on the right side


x=2 or x=-2 Take the square root of both sides

Notice if we graph y=(3x)/(x^2-4), we can visually verify our answers:

drawing(500,500,-10,10,-10,10,<BR> graph(500,500,-10,10,-10,10,(3x)/(x^2-4)),<BR> blue(line(-20,0,20,0)),<BR> green(line(2,-20,2,20)),<BR> green(line(-2,-20,-2,20))<BR> ) Graph of y=(3x)/(x^2-4)) with the horizontal asymptote y=0 (blue line) and the vertical asymptotes x=2 and x=-2 (green lines)