Is there a systematic way to find the range of functions? 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.

If you want to go by fixed steps, you should do a proper analysis of the function. Back in the days at my school that meant the following:
1) State Domain
2) Investigate y-intercept
3) Investigate x intercept(s)
4) Investigate when the curve is above/under x-axis
5) Investigate vertical asymptotes/holes in the graph
6) State derivative
7) State Domain derivative
8) Make a numberline and determine intervals when function is increasing/decreasing
9) Determine max/min
10) Consider limits $x$ goes to infinity's, horizontal asymptotes?
11) Possibility of slant asymptotes or other asymptotic behavior.
12) Make a graph (nowadays use a calculator)
13) Determine Range
An option would be second derivative but in my school that was not standard on the list unless specified by the teacher.
Hope this helps
