I have random variables $X$, $Y$ with joint distribution $f_{XY}(x,y)$ and conditional distribution $f_{X|Y}(x|y)$ and another random variable $Z=g(X)$ with $g$ being bijective is it true that
$$E(Z|Y=y)=\int_{-\infty}^{\infty}g(x)f_{X|Y}(x|y)dx$$
if so, does $g$ need to be bijective for this to hold in general? If not, is there a way to find $E(Z|Y=y)$ knowing just the joint and conditional probability functions for $X$ and $Y$?