Show Independence of two random variables $X$ and$Y$ Suppose that $X$ and $Y$ are independent random variables and $g$ is a real-valued function on $R$.
Show that $g(X)$and $Y$ are independent.
Okay, so i dont really know where to begin with this to be honest,
What i do know is that if $X$ and $Y$ are independent random variables then $$E(XY)+EXEY$$ and $$var(X+Y)=varX+varY$$
Would this information help me with my answer? Would i use one of the probability theorems for independence such as $P(A\cap B)=P(A)P(B)$?
EDIT:
Would i be looking to find $$\mathbf{P}(g(X) = x, Y=y) = \mathbf{P}(g(X) = x) \cdot \mathbf{P}(Y=y),$$
to show independence?
If so what how would i go about this? Many thanks
Any help much appreciated, many thanks 
 A: Assume that $X$ and $Y$ are independent, that is
$$
P(X\in A,Y\in B)=P(X\in A)P(Y\in B),\quad \text{for all}\; A,B\subseteq\mathbb{R}.\tag{1}
$$
To show that $g(X)$ and $Y$ are independent, we have to show $(1)$ with $X$ replaced by $g(X)$.
Let $A,B\subseteq\mathbb{R}$. Since $g:\mathbb{R}\to\mathbb{R}$, we have that $g^{-1}(A)=\{x\in \mathbb{R}\mid g(x)\in A\}$ is also a subset of $\mathbb{R}$. Thus
$$
\begin{align}
P(g(X)\in A,Y\in B )&=P(X\in g^{-1}(A),Y\in B)=P(X\in g^{-1}(A))P(Y\in B)\\
&=P(g(X)\in A)P(Y\in B)
\end{align}
$$
and hence $g(X)$ and $Y$ are independent.
Note that in general we can only talk about the probability of $P(X\in A)$ for certain nice subsets $A$. These are called Borel sets, and hence $(1)$ is only required to hold for Borel sets $A$ and $B$. But this is probably out of the scope of your question.
A: If the joint density function of $X$ and $Y$ is the product of the individual densities then they are considered independent. Therefore, $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ the $X$ and $Y$ are independent. Let $Z=f(x)$. You have to show that:
$$f_{Z,Y}(z,y) = f_Z(z) f(y)$$
Start from the joint CDF of $Z$ and $Y$ and see if you can work it out.
