Let $X$ be a random variable, and let $\phi\_X(t)$ be its characteristic function. Let $Y = f(X)$ be a transformation of the random variable $X$ where $f$ is increasing and one-to-one. Is there a direct functional relationship between the characteristic function of the transformation $f(X)$ and the characteristic function $\phi\_X(x)$? Meaning if $\phi\_Y(t)$ is the characteristic function of the random variable $f(X)$, can we write $\phi\_Y(t)$ in terms of $\phi\_X(t)$?