real life usages of (x+abs(x))/2

Question

I know for all $x<=0$ that $y=0$ and that for all $x>=0$ that $y=x$. I have been wondering if there are any real-life uses of this equation

Answer 1

From the way you have worded it I presume you mean: $$y=\frac{x+|x|}{2}$$ is the same as: $$y=0,\,x\le0$$ $$y=x,\,x>0$$ This is similar in ways to how the heaviside step function would be used, as there are certain situations where an effect is only required after a given time.

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E.g. If I have a varying force applied modelled by $\sin(t)$ for $t\ge0$ and then at a certain point, $t_0$ an additional increasing force is applied, this could be modelled as: $$F=\sin(t)+\frac{(t-t_0)+|t-t_0|}{2}$$

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Thanks to @peterwhy's comments which highlighted another use situation which I forgot which is Macaulay notation. This often takes the form $\langle x\rangle$ and is used in mechanics when looking at forces applied along beams. There is a whole article about Macaulay's method here