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.

  • $\begingroup$ 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 '15 at 1:59
  • $\begingroup$ @imranfat: So, there is no fixed steps to follow ? $\endgroup$ – Mohamed Mostafa Sep 12 '15 at 11:24

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

  • $\begingroup$ So, there is no direct way for this target except making full analysis. Thanks a lot. $\endgroup$ – Mohamed Mostafa Sep 12 '15 at 23:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.