what is "one-to-one reduction from a function f to another function g" I am reading a paper called "Rational Proof". It mentioned the following one-to-one reduction. I cannot google an introduction of it. 

An excerpt from the paper.
"Recall that a one-to-one reduction from a function $f$ to another function $g$ is a triple of polynomial time computable functions ($\alpha$, $\beta$, $\gamma$) such that:


For all $x$ in the domain of $f$ we have $y=f(x)$ if and only if $g(\alpha(x))=\beta(y)$
For all $x$ in the domain of $f$, let $w=\alpha(x)$. Then we have $g(w)=z$ if and only if $f(x)=\gamma(z)$.
"


Any reference for the formal definiton of the so-called "one-to-one reduction" is appreciated. Thanks.
The definition is composed of two parts. First, we want a reduction from $f$ to $g$. That's a way of computing $f$ given an oracle to $g$. This reduction is formed by two polytime functions $\alpha,\gamma$ which are used in the following way: $f(x) = \gamma(g(\alpha(x))$.

Second, we want the reduction to be one-to-one. That explains the if-and-only-if in (2). Moreover, for some reason we want an efficient one-to-one right-inverse $\beta$ of $\gamma$, satisfying $y = \gamma(\beta(y))$. The function $\beta$ can be used to answer the following question: what value of $g$ corresponds to a given value of $f$? (In contrast, $\gamma$ tells us what value of $f$ corresponds to a given value of $g$.) Why this is need will be apparent from the way the paper uses $\beta$.

The actual properties stated in the definition are slightly weaker (you want the function to be one-to-one only for values which actually occur in the formulas), but this is the gist of it.