Domain and range of a multiple non-connected lines from a function? How do you find the domain and range of a function that has multiple non-connected lines? 
Such as, $ f(x)=\sqrt{x^2-1}$. Its graph looks like this:


I'm wanting how you would write this with a set eg: $(-\infty, \infty)$.
P.S. help me out with the title. Not sure how to describe this.

The domain of your function includes all values of $x$ for which the function is defined: 
$$f(x)=\sqrt{x^2-1}$$
is defined if and only if $\quad x^2 - 1 = (x+1)(x-1) \geq 0$. 
So we need to rule out all values for which $\;x^2 -1 < 0$, which happens when and only when $|x| < 1$, and that happens for any $x$ is in the interval $ (-1, 1)$. So we need to exclude the open interval $(-1, 1)$ from the domain of $f$, giving us a domain of $f$ all of $\mathbb R\setminus (-1,1)$, where $f$ is defined.


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*So that gives us that the domain of $f$ is equal to $(-\infty, -1] \cup [1, +\infty)$.


As you can see by your graph, the range of $f(x)$ includes all values $y = \sqrt{x^2 - 1} \geq 0$. 


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*Hence the range of the function lies withing the interval  $[0, +\infty)$


Your graph is "off" a bit, on the $x$-intercepts: the two curves you see should intersect the x-axis at the points $(-1, 0)$ and $(1, 0)$. Here's a zoomed-in graph of  the function near the origin:
$f(x) = \sqrt{x^2 - 1}$

And here it the same function zoomed "out"

