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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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  • $\begingroup$ This is a very broad question though. It really depends on the difficulty of the function. Generally I would say: First graph. That's why we have graphing calculators these days. Once the graph shows some interesting features regarding max/min and asymptotic behavior, we can (or must) resort to calculus to confirm what the graph is showing. From there we ought to be able to establish a Range $\endgroup$
    – imranfat
    Sep 12, 2015 at 1:59
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    $\begingroup$ @imranfat: So, there is no fixed steps to follow ? $\endgroup$ Sep 12, 2015 at 11:24

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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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  • $\begingroup$ So, there is no direct way for this target except making full analysis. Thanks a lot. $\endgroup$ Sep 12, 2015 at 23:04

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