SOLUTION: Given {{{y=(2x^2-18)/(x^2-25)}}}, A) Find the domain. B) Determine the vertical asymptote(s). C) Determine the horizontal asymptote or oblique asymptote. D) Find th

Question 149237: Given ,

A) Find the domain.
B) Determine the vertical asymptote(s).
C) Determine the horizontal asymptote or oblique asymptote.
D) Find the y-intercept.
E) Find the x-intercept(s).
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
A)
Domain:

Start with the given function

Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain.

Factor the left side (note: if you need help with factoring, check out this solver)

Now set each factor equal to zero:

or

or Now solve for x in each case

So our solutions are or

Since and make the denominator equal to zero, this means we must exclude and from our domain

So our domain is:

which in plain English reads: x is the set of all real numbers except or

So our domain looks like this in interval notation

note: remember, the parenthesis excludes -5 and 5 from the domain

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

Set the denominator equal to zero

Add 25 to both sides

Combine like terms on the right side

Take the square root of both sides

or Simplify

So the vertical asymptotes are or

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Looking at the numerator , we can see that the degree is since the highest exponent of the numerator is . For the denominator , we can see that the degree is since the highest exponent of the denominator is .

C)
Horizontal/Oblique 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 , the leading coefficient is

Looking at the denominator , the leading coefficient is

So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide by to get

So the horizontal asymptote is

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D)
Y-Intercept:

To find the y-intercept, simply plug in

Start with the given function

Plug in

Simplify

So the y-intercept is

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E)
X-Intercept(s):

To find the x-intercept(s), simply plug in and solve for x

Start with the given function

Plug in

Since the denominator cannot be equal to zero, this means that the numerator is equal to zero.

Add 18 to both sides.

Divide both sides by 2.

or Take the square root of both sides.

So the x-intercepts are

and

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Notice if we graph , we can visually verify our answers:

Graph of with the horizontal asymptote (blue line) and the vertical asymptotes and (green lines)

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