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.
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$\begingroup$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
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