How do you find the domain and range of #y=(2x)/(x+9)#?
I know this is an extremely long answer, but hear me out.
First, to find the domain of a function, we must take note of any discontinuities that occur. In other words, we have to find impossibilities in the function. Most of the time, this will take the form of
Removable discontinuities are "holes" in the graph that are just a sudden break in the line, interrupting only one point. They are identified by a factor being present in both the numerator and denominator. For example, in the function
we can use the difference of squares to determine that
Here we now can observe that there is a factor of
Non removable discontinuities create vertical asymptotes in the graph that interrupt the points before and after the point that doesn't exist. This what the equation you stated concerns. In order to determine the location of such asymptotes. We will have to find any values of
Using basic algebra, we can determine that the in order for the denominator to equal 0,
After finding all types of discontinuities in the graph, we can write our domain around them using our friend, the union sign:
For determining the range of the function, there are three rules which describe the end behavior of functions. However, there is one that applies to yours, it is, in a more casual way:
If the largest powers of the variables in the numerator and denominator are equal, then there is an asymptote at
In terms of your equation, the powers of your largest power variables are equal, so I divide the coefficients of 2 and 1 to get