SOLUTION: Let f(x)=2x^2+7x-4/x^3-1 A. What is the domain of f(x)? B.What is the range of f(x)? C.What are the roots of f(x)? D.determine the end behavior of f(x)? E.Where will

Question 935794: Let f(x)=2x^2+7x-4/x^3-1
A. What is the domain of f(x)?
B.What is the range of f(x)?
C.What are the roots of f(x)?
D.determine the end behavior of f(x)?
E.Where will f(x) intersect the y-axis?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
You wrote 2x^2+7x-4/x^3-1 = ,
but I suspect that you meant f(x)=(2x^2+7x-4)/(x^3-1)= .
If ,
A. the domain of is all the values of that give a value to .
That means all the values of that
make , and that is all the values of such that
,
because --->---> .
Other that for , is a continuous function, so it varies continuously.
It is zero and it changes sign at the points where .
It has no absolute maximum or minimum, because
as approaches ,
the numerator approaches ,
while the denominator approaches ,
which makes the absolute value of to increase without bounds.
At the denominator changes sign,
with <---> and <---> ,
so ---> as approaches from the left, and ---> as approaches from the right.

B. The range of is all the values can take.
That is all real numbers or

.
Graphing the function would show you that,
but I am not sure how you are expected to demonstrate that.
One way would be to demonstrate that
<---><---> has a solution for every possible real value of .
It is obvious that it is so,
because is an odd-degree polynomial,
an those always hit zero somewhere.

C. The roots of are the roots of ,
and we find them by solving .
Factoring we get
-->-->--> .
So, the roots are and .

D. The end behavior of is given by
.
The limit is so because rational functions tend to zero at the ends
when the degree of the denominator (3 in this case)
is greater than the degree of the denominator (2 in this case).
So, (the x- axis) is the horizontal asymptote.
The function is positive for ,
so it approaches the positive x-axis from above.
At the other end, where ,
approaches the negative x-axis from below.

E.Since the y-axis is the line ,
f(x) intersect the y-axis at the point with , (0,4).
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