Question

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

Answer 1

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`.