Q.Find the domain and range of the rational function $f(x) = \dfrac{1}{x-2}$.

A rational function is undefined wherever its denominator is zero, and its range can be found by solving for $x$ in terms of $y$.

$f(x)$ is undefined at $x=2$ since division by zero is not defined, and $y=0$ is never attained since $\dfrac{1}{x-2}$ can never equal zero for any real $x$.

Finding the domain: $f(x) = \dfrac{1}{x-2}$ is defined for every real $x$ except where the denominator becomes zero:

So $f$ is undefined only at $x=2$, and the domain is

Finding the range: let $y = f(x) = \dfrac{1}{x-2}$. Solve this for $x$ in terms of $y$:

This shows that for every real $y$ except $y=0$, there is a valid real $x$ satisfying $f(x)=y$; but $y=0$ itself is impossible, because $\dfrac{1}{x-2}=0$ would require the numerator $1$ to equal $0$, which it never does. So the range is

A common error is stopping after finding the domain and forgetting to work out the range separately, or assuming the range excludes the same number ($2$) that the domain excludes — the excluded value of $y$ is found by solving for $x$, not by copying the excluded value of $x$.