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

1 Answer
Zor Shekhtman
May 28, 2016

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.

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