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Question 635876: Please help me solve this problem : What is the domain,the range,the intercepts,the symmetries and asymptotes of y=x+2/x-2?
thank you
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Please help me solve this problem :
y = (x+2)/(x-2)
What is
the domain:::all Real numbers except x = 2
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the range:: all Real numbers except y = 1
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the intercepts :::
x-intercept: (-2,0)
y-intercept: (0,-1)
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the symmetries:::
f(-x) = (-x+2)/(-x-2) is not equal to f(x)
-f(-x) = (x-2)/(-x-2) is not equal to f(x)
No y-axis or origin symmetry
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and asymptotes:::
Vertical Asymptote: x = 2
Horizontal asymptote: y = 1
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Cheers,
Stan H.
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
The domain of a function is that set of values for which the function is defined. In the case of a rational function like this one, the domain is all real numbers EXCEPT any number that would make the denominator equal to zero. Set the denominator equal to zero and solve to find the value(s) that must be excluded from the domain.
The range is the set of values that the function can become. In this case if  decreases without bound, then the value of the function will increase toward the value 1, but it will never actually be exactly 1. You can make the value of the function as close to 1 as you like by chosing a sufficiently large negative value for . Same thing on the other side when increases without bound, except that we get close to 1 from above, the "greater than 1" side. So far we have excluded the value 1 from the range. As x, starting less than the number you determined should be excluded from the domain, gets close to that number, the value of the function is going to take off in the negative direction, and you can make the value of the function as large a negative number as you like by moving closer to that value that you excluded from the domain. Likewise, when approaches the domain excluded value from the right side, the value of the function takes off in a positive direction, and you can make the value of the function as large as you like by moving closer to that excluded value. So we have demonstrated that the range of this function is all real numbers excluding 1.
The  -intercept is the ordered pair \right)) . The intercept is found by finding the zero(s) of the function. In this case, simply set the numerator of the function equal to zero and solve for the value  . Then ) is the -intercept.
Substitute  for in ) and simplify. If \ =\ f(-x)) , then the function has even symmetry. Substitute for in and then multiply the result times -1. If \ =\ -f(x)) , then the function has odd symmetry. If neither, then the function has no symmetry.
The function has a vertical asymptote at the line  for every value  that is excluded from the domain except those values that are also zeros of the function. A rational function where the degree of the numerator polynomial is equal to the degree of the denominator polynomial has a horizontal asymptote at the line  where  is the lead coefficient of the numerator and  is the lead coefficient of the denominator.
The only thing left for you to do is the arithmetic.
John
My calculator said it, I believe it, that settles it
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