The graph of symmetric-decreasing rearrangment of some function eg. $e^{x}$ $A^*$ to be the ball centered at 0 with the same measure that

$A$.
  The symmetric-decreasing rearrangement of a measurable function $f:\mathbb{R}^n \to \mathbb{R}$ is then defined by
   $$f^*(x):=\int_0^{\infty} \chi_{\{|f(x)|>t\}^*}(x)dt,$$

by comparison to the "layercake" representation of $f$, namely
 $$f(x)=\int_0^{\infty} \chi_{\{f(x)>t\}}(x)dt.$$
So is there a way I can get a formula out of $f^{*}$ for some examples (eg. $e^{x}$), so I can graph them. I know they will all look like parabolas but I would like to know of any precise ways to see them (maybe computational-Mathematica).
thanks
 A: The symmetric decreasing rearrangement is not defined for $e^x$ on $\mathbb R$. The definition requires $|\{|f|>t\}|<\infty$ for all $t>0$. Also, it's better to just assume $f\ge 0$ since we disregard the sign anyway.
For a randomly picked function like $f(x)=x^2 e^{-x^2}$, the rearrangement is unlikely to have a closed form, because it is tied to solving the transcendental equation $f(x)=t$ for $x$. For doable and illustrative examples, it's best to pick algebraic functions on a finite interval (define them to be $0$ outside of the interval). 
Example 1. $f(x)= x$ on $[0,10]$, zero elsewhere. Then $f^*(x)= 10-2|x|$ on $[- 5,5]$, zero elsewhere. 
Example 2. $f(x)= x^2$ on $[0,10]$, zero elsewhere. Then $f^*(x)= (10-2|x|)^2$ on $[- 5,5]$, zero elsewhere. Which illustrates a point: $(\phi\circ f)^*=\phi\circ f^*$ when $\phi$ is increasing on the range of $f$. 
Example 3, non-monotone. $f(x)=|x|$ on $[-1,3]$, zero elsewhere. Then $f^*(x)=3-2|x|$ on $[-1,1]$, $f^*(x)=2-|x|$ when $1\le |x|\le 2$, and zero elsewhere. 
Example 4, also non-monotone. $f(x)= x^4+3x^2$ on $[-1,3]$, zero elsewhere. Looks hard? No, just apply the observation after example 2 to Example 3.
