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Question 1027061: For the function: (x^3+2x^2-4x-8)/2x^2-8
a) Find the domain of the function, and discuss the behavior of f near any excluded x values.
b) Identify all intercepts
c) Find all vertical, horizontal, and slant asymptotes of the graph of the function.
Found 2 solutions by robertb, KMST:
Answer by robertb(5830)   (Show Source):
Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website! I assume you meant
 .
The numerator and denominator of that rational function can be factored.
 .
When  <---> the denominator is zero,
and the function has no defined value.
a) So, the values  are excluded from the domain of the function.
The function can be simplified as
 for  .
The graph of  is a straight line with two holes,
and the behavior of near the excluded values (and everywhere else in its domain)
is just like the behavior of  .
In other words,  and  .
b) The y-intercept of is  ,
or the point (0,1) .
is never zero, because where the function exists
it is equal to  ,
and  only for  which is not part of the domain of .
So, there is no x-intercept. The graph of function does not intersect the x-axis; it jumps over it.
c) Since everywhere in its domain  ,
the line  could be considered the slant asymptote. There is no other asymptote.
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