Final answer:
The oblique asymptote of the function f(x)=(x³ -x²+x-1)/(x² +3x-2) is the line y=x-4. This is found by performing polynomial long division of the numerator by the denominator.
Step-by-step explanation:
The student is asking to determine the oblique asymptote for the function f(x)=(x³ -x²+x-1)/(x² +3x-2). To find the oblique asymptote of a rational function, we look for where the degree of the numerator is one higher than the degree of the denominator. In this case, we must divide the numerator by the denominator to obtain the asymptote.
We perform polynomial long division or synthetic division to simplify f(x).
Since the degrees of the polynomials fulfill the condition for an oblique asymptote, we expect there to be a linear term remaining after division.
After dividing x³ -x²+x-1 by x² +3x-2, the quotient is x-4 with a remainder.
This quotient x-4 represents the oblique asymptote of the given function, and the graph of the function f(x) will approach the line y=x-4 as x tends towards positive or negative infinity.