How to find the range of a function without a graph? Okay so I know how to find the domain of the function  $f(x) = \frac{3x^2}{x^2-1}$, which is $x\neq 1$ and $x\neq -1$, but I'm totally confused on how to find the range without using a graph.

Write the function as 
$$f(x)=\frac{3x^2-3+3}{x^2-1}=3+\frac3{x^2-1}$$
As $f$ is even, we may suppose $x\ge 0,\enspace x\ne q 1$.
Now the range of $x^2-1$ is $[-1,+\infty)$ and for $f(x)$, the value $x^2=1$ is excluded, hence the range of $\;\dfrac3{x^2-1}$ is $(-\infty,-3] $ for $x\in [0,1)$ and $(0,+\infty)$ for $x>1$.
Hence the range of $f(x)=3+\dfrac3{x^2-1}$ is $\;(-\infty,0]\cup (3,+\infty)$.
