I was looking for steps or a systematic way to find the range of functions.
The way to find domain is quite obvious: exclude x-values that make the function undefined on the real numbers.
But the range is harder, I found several methods to find range:
$1)$ Intuition: to guess how the function behaves.
$2)$ Graph: to graph the function and get the range from it.
$3)$ Using limits and calculus to determine the min and max, and how the function behave at infinity.
$4)$ Domain of Inverse of function.
But the problem is there are some functions really hard to get their inverses. And I want to find the range analytically without graphs, limits or calculus tools.
Is there any systematic and direct way to find the range of functions ?
Thanks a lot for help.