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$\begingroup$

A user on the chat asked how could he make something that would cap when it gets a specific value like 20. Then the behavior would be as follows:

$f(...)=...$

$f(18)=18$

$f(19)=19$

$f(20)=20$

$f(21)=20$

$f(22)=20$

$f(...)=20$

He said he would like to perform it with a regular calculator. Is it possible to do this?

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    Could you better explain your problem?2012-08-20
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    Well, I guess you are not looking for the function $f(x)$ which returns $x$ when $x$ is less than 20, and 20 otherwise.2012-08-20
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    Your use of "=" has so many people rolling in their graves now I think you have triggered the mathematician zombie apocalypse. Do you mean something like capping a sequence $f(1),f(2),f(3)\dots$?2012-08-20
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    From [here](http://math.stackexchange.com/questions/13237/): $$\frac{x+20-|x-20|}{2}$$2012-08-20
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    @utdiscant What is this function?2012-08-20
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    Calculators might have $min(x,20)$?2012-08-20
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    @rschwieb 1 - Why the problem with the $=$? 2 - Yep, that's what I want.2012-08-20
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    @GustavoBandeira it belongs to a Piecewise function. Looks something like [this](http://en.wikipedia.org/wiki/Piecewise_function#Notation_and_interpretation).2012-08-20
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    $f(x) = \min(x,20)$2012-08-20
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    @FrenzYDT. That's a problem. Piecewise functions work with conditionals, he wants it without conditionals.2012-08-20
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    Should we add a [tag:calculator] tag?2012-08-20
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    @GustavoBandeira You shouldn't put "=" between quantities that aren't equal. If you do, you have a false statement. Writing "22=20" is as absurd as writing "0=1".2012-08-20
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    @rschwieb I like to be experimental, dude.2012-08-20
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    @GustavoBandeira Quick fact: misunderstanding the equals sign cripples the math ability and development of students at many levels (no kidding). A good portion of them (mistakenly) think that this problem "isn't important". Take it seriously!2012-08-20
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    @rschwieb I take it seriously. But at this context, I needed something to express this. I guess it would be better to: $f(19)=20$, $f(20)=20$, $f(21)=20$, $...$2012-08-20
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    @GustavoBandeira Yeah, that's a much better version :) People will understand you more quickly.2012-08-20
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    @J.M. You should give it as a answer because it is the answer. I've noticed it won't work for the negative numbers, is it possible to make a more general form including the integers?2012-08-20
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    @Gustavo: It seems to me that J.M.'s formula works for all numbers. What makes you think it wouldn't for negative numbers?2012-08-20
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    Great, not only is this turning into an obfuscated formula competition, but people are ignoring the part where "he said he would like to perform it with a regular calculator."2012-08-20
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    Of course if you chose a moderately large, but finite $n$ ($n=10$ is already quite good), my formula can be used on a standard scientific calculator. While it only gives an approximation, only for values near $x=20$ the approximation differs from the exact value less than the numeric error. However, for too small $x$ you may get an overflow.2012-08-21

3 Answers 3

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$ x \mapsto \min ( x , 20 ) $

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We can also get a bit (unnecessarily) fancier: $$ f(x) = x + (20 - x) \int\limits_{-\infty}^{x-20} \delta(t)\ dt $$ where $$ \int\limits_{-\infty}^{x-20} \delta(t)\ dt = \begin{cases} 0 & x < 20 \\ 1 & x \ge 20 \end{cases} $$ (See Heaviside step function.)

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While we are at fancy expressions, what about $$20-\lim_{n\to\infty}\frac1n\ln\left(1+\mathrm e^{n(20-x)}\right)$$