How to find Domain, Range and Asymptote?

Question

How to find the Domain, Range and Asymptotes of the following Rational Function?

 

y = 2x-6 / x-6

Answer 1

Well, here is a quick definitions to remind you. Domain is the set of inputs, in other words, all possible x and Range is the set of outputs or y values resulted from those x. Clearly, in this graph, x can't be 6 because the denominator x-6 can't be 0. And any other values of x is possible, you can double check. Then, the Domain is the set of real number, but 6 exclusive.

Now, for range, it "seems" like y can be any real numbers, but if you multiply with (x-6) to both sides, you get 

y(x-6) = 2x-6. If y was 2, the left side would be 2x-12 = 2x-6 which is absurdly wrong and no solution, meaning y can't be 2, but any other real numbers. Therefore, the Range is the set of real numbers, 2 exclusive. Asymptotes are basically a point where the graph would approach closer and closer, but will never meet the point in any finite distance. In this case, the graph will never meet x = 6 and y = 2 which are vertical asymptote and horizontal asymptote respectively.