Question

I am given the rational function:
f(x)= (x^3 - 1) / (x^2 - 4)
I found that the domain is all real numbers except x= plus or minus 2 , I also found that the x intercept is x=1, however I do not get how to find:
the y-intercet , or identify the Horizontal, vertical or slant asymptotes.
I got that there was no horizontal asymptotes, but I am more confused on the V.A and slant and y intercept.
Please help me

Answer 1

surely you realize that the y-intercept is where the graph crosses the y-axis!?! That is, where x=0...

f(0) = (-1)/(-4) = 1/4

Since the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote.

A simple division shows that

f(x) = x + (4x-1)/(x^2-4)
or,
f(x) = x + (7/4)/(x-2) + (9/4)/(x+2)

In either case, as x gets huge, the fractions go to zero, so the slant asymptote is y=x

For vertical asymptotes, you need the denominator to be zero, and not the numerator. The reason the domain excludes x = ±2 is that that's where the vertical asymptotes are!