Let $X,Y$ be continuous random variables with $X$ defined over $\mathcal{A}$, and let $f: \mathcal{A} \to \mathcal{A}$, $g: \mathcal{A} \to \mathcal{A}$ be any functions. Is it true that

$$
I(Y;f \circ g(X) ) \leq I(Y; f(X))
$$

where $I(\cdot\,; \cdot)$ denotes mutual information. Note that
data processing inequality gives,
$$
I(Y; f\circ g(X)) \leq I(Y; g(X)) \leq I(Y; X)\\
I(Y; f(X)) \leq I(Y;X)
$$