for the equation y=root 4-x^2
how do you find the range and domain???
for the equation x^2-9x+20 / x-5
how do you find the range???
Generally, the most effective method for finding the domain is to assume "All Real Numbers" and then see if you can rule out anything.
In the first, 4 - x^2 needs to be positive. Can you determine where it is negative?
In the first, x^2 is always positive. What's the greatest value that 4 - x^2 can take?
You must think about these. There is not a magic formula.
Minor nitpick, but the OP did not ask for the domain, he/she asked for the range. Nevertheless, your point still holds, the notion of canceling the "x-5" factors. However, knowing that the graph of the original f(x) is almost the same as g(x) = x - 4 (I'm calling this g(x)) is helpful to find the range. The graph of both f(x) and g(x) are lines, except for at x = 5. At x = 5, f(x) is not defined but g(x) = 1. That tells me that the range of f(x) is $\displaystyle ( -\infty, 1) \cup (1, \infty)$.