How do you find the domain and range of #f(x) = (2x)/(x-3) #?

1 Answer
May 28, 2016

Answer:

Domain of function #f(x)=(2x)/(x-3)# is all real numbers except #3#.

Range is all real numbers except #2#.

Explanation:

Graph of #f(x)=(2x)/(x-3)# is

graph{2x/(x-3) [-20, 20, -10, 10]}

Domain, the set of argument values where the function is defined, is, obviously all real numbers except those where denominator #(x-3)# is zero, and it happened to be only #x=3#. So, for all #x!=3# the function is defined and its domain is #x!=3#.

One of the ways to determine the range of a function #f(x)# is to consider a domain of an inverse function #f^(-1)(x)#. In our case, if #y=(2x)/(x-3)# then #x=(3y)/(y-2)#. Therefore, inverse function is #f^(-1)=(3x)/(x-2)#, and its domain is #x!=2#. So, the range of the original function is all real numbers except #2#.

Actually, the number #2# is the limit our function is approaching as #x# tends to #+oo# or #-oo#.