$($a$)$ Let $0$; define a function $S_{}(x)$ as follows:

$S_{}(x)$:

For this function $S_{}(x),$ by using the function $(x{)}_{+}=$max$\{x,0\}$ and sign$(x)=\{\begin{array}{cc}-1,& x<mn>0</mn>\\ 0,& x=0\\ 1,& x>0\end{array}$

show that it can be re$-$expressed as $S_{}(x)=$sign$(x)(|x|-{)}_{+}.$

$($b$)$ Write a program in Python that performs $($a$).$ Name it

soft.th$.$ Then, exe$-$

cute the following code to see whether it works correctly.

x$\}=$np$.$arange$(-10,10,0\mathrm{.}1$

y $=$ soft$\_$th$(5,$x$)$

plt$.$plot$($x$,$ y$,$ c$=$"black"$)$

plt$.$title$($r$={?}_S{?}_{}(x)"*,$ size$=24)$

plt$.$plot$([-5,-5],[-4,4],$ c$=$"blue", linestyle$=$"dashed"$)$

plt$.$plot$([5,5],[-4,4],$ c$=$"blue", linestyle$=$"dashed"$)$

plt$.$text$(-2,1,$ r$"=5",$ c$=$"red", size$=16)$

Hint In the R language, if we define a function by dividing it into cases or

using max, it will not plot a graph for us$.$ Use sign, abs$,$ and pmax instead.