Distribution of Functions of Random Variables In general, how would one find the distribution of $f(X)$ where $X$ is a random variable? Or consider the inverse problem of finding the distribution of $X$ given the distribution of $f(X)$. For example, what is the distribution of $\max(X_1, X_2, X_3)$ if $X_1, X_2$ and $X_3$ have the same distribution? Likewise, if one is given the distribution of $ Y = \log X$, then the distribution of $X$ is deduced by looking at $\text{exp}(Y)$?

You can use the law of conditional probability:
$P(A)=\int^{\infty}_{-\infty}P(A|B)f(B)$
So in your case, for a random variable $X\in[0,1)$:
$P(x>f(X))=\int^{\infty}_{-\infty}[x>f(a)][0<a<1]da=\int^{1}_{0}[x>f(a)]da$
Where the brackets are Iverson brackets.
Example, the distribution for a random variable $X\in[0,1)$ squared:
$P(x>X^2)=\int^{1}_{0}[x>a^2]da=\int^{1}_{0}[\sqrt{x}>a]da=\int^{\sqrt{x}}_{0}1da=\sqrt{x}$
Assuming $x\in[0,1]$. The pdf is then $\frac{d\sqrt{x}}{dx}=\frac{1}{2\sqrt{x}}$.
