Domain and Range of a Rational Function — Cheat sheet

Key method

To find the range, solve $y=p(x)/q(x)$ for $x$, then check for impossible targets and forbidden solutions. Removable holes can exclude an otherwise attainable output.

Example

For $f(x)=(x+1)/(x-2)$, the domain excludes $2$. Solve $y(x-2)=x+1$:

This expression never equals $2$, so every $y\ne1$ is attained. At $y=1$, the equation would say $-2=1$, impossible. The range excludes $1$.

Avoid this mistake

For $(x^2-1)/(x-1)$, canceling gives $x+1$ only when $x\ne1$. The hole means the range also excludes $2$.