SOLUTION: 5. Determine the behavior of {{{f(x)=(2x^3-4x^2-16x)/(x^3-4x)}}} Domain : x � intercept(s) : Hole(s) : y � intercept : Vertical Asymptote(s) :

Question 113820: 5. Determine the behavior of
Domain :
x � intercept(s) :
Hole(s) :
y � intercept :
Vertical Asymptote(s) :
Horizontal Asymptote :

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Domain: Simply set the denominator equal to zero and solve for x

Start with the given equation

Factor out an x

Factor using the difference of squares

, , or Set each factor equal to zero

, , or Solve for x in each case

Since , , or makes the denominator zero, we must take these values out of the domain.

So our domain is:

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

x-intercept(s): To find the x-intercepts, let f(x)=0 and solve for x

Start with the given equation

Set the numerator equal to zero. Remember the denominator cannot equal zero. So the numerator
can only equal zero.

Factor out 2x

Factor to

, , or Set each factor equal to zero

, , or Solve for x in each case

So the possible x-intercepts are: , , or

However, since x=-2 and x=0 is not in the domain, the only x-intercept is

Answer: The x-intercept is

Holes:

Start with the given function

Factor out an x out of the denominator

Cancel like terms

Simplify

Factor the numerator and denominator

Cancel like terms

Simplify

So here we can see that is equivalent to . However, we must
keep in mind to exclude x=-2, x=0, and x=2 (remember the domain). Since we can plug in x=-2 and x=0 into
(but not into ) we must make a mental note that there are holes
at x=-2 and x=0.

Answer: holes at x=-2 and x=0

y-intercept(s): To find the y-intercepts, let x=0 and solve for y

Start with the simplified equation

Plug in x=0

Combine like terms

Multiply

Multiply

So when x=0, f(x) (which is y) is equal to 4.

Answer: So the y intercept is (0,4)

Vertical Asymptote(s):

Vertical asymptotes occur when the denominator is equal to zero. In other words, at the value of x that is not in the domain.

Start with the simplified equation

Answer: So the vertical asymptote is

Horizontal Asymptote(s):

To find the horizontal asymptote(s), simply take the leading coefficient of the numerator (which is 2) and divide it by the leading coefficient of the denominator (which is 1)

So simply divide 2 by 1 to get

So the horizontal asymptote is

If we graph the function , we can visually verify all of our answers:

Graph of

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