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October 3rd, 2015, 07:08 AM   #1
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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?
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October 3rd, 2015, 09:07 AM   #2
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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).
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Last edited by LearnMathsFree; October 3rd, 2015 at 09:11 AM.
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October 3rd, 2015, 11:17 AM   #3
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Or $y = x\sqrt{x^2}$, a form that's convenient if you wish to find $dy/dx$.
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October 6th, 2015, 01:28 PM   #4
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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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