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

May 28, 2016

Domain of function $f \left(x\right) = \frac{2 x}{x - 3}$ is all real numbers except $3$.

Range is all real numbers except $2$.

#### Explanation:

Graph of $f \left(x\right) = \frac{2 x}{x - 3}$ 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 $\left(x - 3\right)$ is zero, and it happened to be only $x = 3$. So, for all $x \ne 3$ the function is defined and its domain is $x \ne 3$.

One of the ways to determine the range of a function $f \left(x\right)$ is to consider a domain of an inverse function ${f}^{- 1} \left(x\right)$. In our case, if $y = \frac{2 x}{x - 3}$ then $x = \frac{3 y}{y - 2}$. Therefore, inverse function is ${f}^{- 1} = \frac{3 x}{x - 2}$, and its domain is $x \ne 2$. 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 $+ \infty$ or $- \infty$.