Given the following rational function: (a) state the domain. (b) find the vertical and horizontal asymptotes, if any. (c) find the oblique asymptotes, if any. & graph. f(x)=x^2+6x-9/x-6

Question

anonymous · Answer

The restriction on the domain for a rational function comes at values of x that cause the denominator to be zero.  In this case, $$dom(f)=\left\{ x|x \neq6 \right\}$$

anonymous · Answer

Vertical asymptotes occur at the points of discontinuity caused by the void in the domain.  So, in this case, the vertical asymptote is at x=6.

anonymous · Answer

horizontal and oblique asymptotes?

anonymous · Answer

No horizontal asymptotes, since the degree of the numerator is bigger than the denominator.  Since it is only bigger by one degree, we get a oblique (sometimes called slant) asymptote at y=x+12.  (I think.... haven't done this kind of problem in a while.)

anonymous · Answer

The equation for the slant asymptote in this case is the quotient of the original function.  The remainder isn't important in this calculation.