I provide my approach in solving this but I amd not entirely sure whether I am correct. Since X~uni(-1,1) $f_X(x)=1/2$ and $F_X(x)=(x+1)/2$.
$F_{X^2}(x)$=$Pr[X^2≤x]$=$2F_X(√x)$=$(√x+1)/2$. Hence $f_{X^2}(x)$=$1/(4√x)$.
Now $f_X(x).f_{X^2}(x)$=$1/(8√x)$. Now what I need is to find $F_{X^2,X}(x)$ which is the part I am struggling to derive. Is $F_{X^2,X}(x)$=$F_{X^2}(x)$ ? I would appreciate any assistance and answers and hope that the steps I have shown are enough. Thanks in advance
It is possible to find the joint distribution of $X$ and $X^2$, but that is the hard way of tackling the problem.
Informally, if we know that $X$ is close to $0$, then $X^2$ cannot be close to $1$. We turn this observation into a formal argument.
It is easy to verify that $\Pr(0\lt X\lt 1/2)\ne 0$. Also, $\Pr(X^2\gt 1/2)\ne 0$. But $$\Pr((0\lt X\lt 1/2)\land (X^2\gt 1/2))=0.$$ Thus $$\Pr((0\lt X\lt 1/2)\land (X^2\gt 1/2))\ne \Pr(0\lt X\lt 1/2)\Pr(X^2\gt 1/2),$$ and therefore our two random variables are not independent.
Remark: Although it is not necessary, we can turn the above into an argument about the joint density of $X$ and $Y=X^2$. You calculated the individual densities. Their product is not $0$ in the rectangle $0\lt x\lt 1/2$, $1/2\lt y\lt 1$. However, there is probability $0$ of $(X,Y)$ landing in that rectangle, so the joint density must be $0$ in that rectangle. Thus the joint density function is not the product of the individual densities.