SOLUTION: Let f(x) = x^4 + 2x/x^2 − 1 . (a) Determine the domain of f. I got x cannot equal 1 and -1 (b) Determine all zeros of f: i cant find this i set the whole expression to zer

Algebra ->  Rational-functions -> SOLUTION: Let f(x) = x^4 + 2x/x^2 − 1 . (a) Determine the domain of f. I got x cannot equal 1 and -1 (b) Determine all zeros of f: i cant find this i set the whole expression to zer      Log On


   

Question 1025032: Let f(x) = x^4 + 2x/x^2 − 1
.
(a) Determine the domain of f. I got x cannot equal 1 and -1
(b) Determine all zeros of f: i cant find this i set the whole expression to zero and multiplied x^2-1 from both sides leaving me with x^4+2x n ow what do i do?
(c) Determine all vertical asymptotes of f: this is x= 1 or x=-1
(d) Determine the y-intercept of f: there is no horizontal asymptotes
(e) Determine the long range behaviour of f? I don't understand this will you be to please help me.
Thank you for all your time!
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Almost assuredly you mean f(x)=(x^4+2x)/(x^2-1).


f%28x%29=%28x%5E4%2B2x%29%2F%28x%5E2-1%29
%28x%28x%5E3%2B2%29%29%2F%28%28x-1%29%28x%2B1%29%29


The numerator and denominator have no factor in common.

Roots and undefined x values will determine intervals on the x-axis. This affects domain.

ZEROS
Look in the numerator.
x is a zero.
also x%5E3%2B2 would be a root.
x%5E3=-2
x=%28-2%29%5E%281%2F3%29 or x=root%283%2C-2%29.
This root or zero is x=-root%283%2C2%29.

VERTICAL ASYMPTOTES
These occur at x=1 and x=-1 because f is not defined for those values. Why? Because the denominator must never become 0.

CRITICAL x VALUES
These are the x values of -root%283%2C2%29, -1, 0, 1.

y-INTERCEPT
Let x be 0. What is y?
f%280%29=%280%5E4%2B2%2A0%29%2F%280%5E2-1%29
f%280%29=0%2F%28-1%29
f%280%29=0.


That should all be very helpful for you, although not everything was completed for you.