Q.Find the natural domain and the range of the rational function $f(x)=\dfrac{2x}{x-4}$.

A rational function is undefined where its denominator is zero; its range is found by solving $y=f(x)$ for $x$.

$f$ is undefined at $x=4$; solving $y=\frac{2x}{x-4}$ for $x$ shows every real $y$ except $y=2$ is attainable, since $y=2$ would require $-8=0$.

Domain. $f(x)=\dfrac{2x}{x-4}$ is defined for all real $x$ except where the denominator is zero:

So the natural domain is $\mathbb{R}-\{4\}$.

Range. Let $y=\dfrac{2x}{x-4}$ and solve for $x$ in terms of $y$:

This gives a valid real $x$ for every $y$ except $y=2$ (which would make the denominator $y-2$ zero). Checking $y=2$ directly: $\dfrac{2x}{x-4}=2 \Rightarrow 2x=2(x-4)=2x-8 \Rightarrow 0=-8$, impossible — so $y=2$ is never attained. Hence

Dual check: taking $y=6$ gives $x=\frac{4(6)}{6-2}=\frac{24}{4}=6$, and indeed $f(6)=\frac{12}{2}=6$. ✓

A common error is copying the excluded $x$-value ($4$) as the excluded $y$-value; the excluded range value must be found separately by solving for $x$. Here it is $y=2$ (the ratio of the leading coefficients), not $y=4$.