Inactive Tutor answered • 01/17/20
So Chris A.,
Domain refers to the possible values for the independent variable ("x" in all of these problems). Every value is possible except when a term in the denominator approaches zero, b/c division by zero is undefined!
(1) So first, you should factor the denominator into a product of terms, if you can.
(2) Next: figure those domain "dropouts", and write the domain pieces in the appropriate way. You know the drill. So for a term in the denominator of (x-n), x = n will be a "dropout", b/c that term would be zero there.
(3) Next: vertical asymptotes occur at domain values wherever you have a "dropout". So state those values, which you determined above in (2). There will be as many as there are "dropout" values -- that's one each for each term involving an x in the denominator, except if "dropouts" are replicate values (such as with x^2, x^3, (x-2)^2, and so on -- these each have a replicate value, due to the exponent).
(4) Last, horizontal asymptotes occur in f(x) as x-> +infinity or x-> -infinity. Because all of these problems have a NET power of x (numerator powers of x minus denominator powers of x) of <0, as x goes to either infinity, f(x) will approach zero. Therefore, zero is a horizontal asymptote for each.
You may encounter more complicated functions as you proceed along in this math unit -- remember that asymptotes can always be related back to what you already know how to do (avoid dividing by zero, anything times zero = 0, when comparing one function to another in a f(x)/g(x) problem drop all polynomial terms but the one with the highest power in each function for calculating the ratio as x->infinity, etc.)
--Cheers, -- Mr. d.
