Stanton D. answered • 01/17/20
Tutor to Pique Your Sciences Interest
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.
