My Math Forum Is It Possible To Write This As A Function With One Definition?

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 October 3rd, 2015, 07:08 AM #1 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 664 Thanks: 87 Is It Possible To Write This As A Function With One Definition? Take this function where x and y will both be negative, both be 0, or both be positive: If x < 0, y = $\displaystyle -x^2$ If x > or = 0, y = $\displaystyle x^2$ Is it possible to write this function with one definition rather than breaking it into two parts?
 October 3rd, 2015, 09:07 AM #2 Member     Joined: Sep 2015 From: London Posts: 31 Thanks: 6 How about $\displaystyle f(x) = x|x|$? $\displaystyle f(x) = x^{2} \text{sgn}(x)$ works too, though I prefer the former (and the two are, in fact, equivalent). Thanks from EvanJ Last edited by LearnMathsFree; October 3rd, 2015 at 09:11 AM.
 October 3rd, 2015, 11:17 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,969 Thanks: 2222 Or $y = x\sqrt{x^2}$, a form that's convenient if you wish to find $dy/dx$. Thanks from EvanJ
 October 6th, 2015, 01:28 PM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 Or use the "Heaviside step function", $H(x)$ which is defined as 0 if $x< 0$, 1 if $x\ge 0$. Your function is $f(x)= -x^2+ H(x)(2x^2)= x^2(2H(x)- 1)$. Last edited by skipjack; October 6th, 2015 at 03:13 PM. Reason: to insert missing 2.

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