Are all functions that take the form $ x ^ 2$ convex (with respect to some parameters)?

Question

I know that $f(x) = x ^ 2$ is a convex function. I can see that from plotting $f(x)$ on 2D graph with respect to $x$.

Now, lets say I have $f(\ldots) = (g(\ldots))^2$ . Let assume $g(\ldots)$ could be a simple linear function or a polynomial or the kind of function whose formula we can't even write it down (eg; type of 'learnt' function with billions of parameter commonly found in deep neural network).

I assume function "$f$" is still convex when plotted with respect to '$g$'?

But will '$f$' still be considered convex when plotted with respect to some (let's say 2 parameters) of a billion parameters of '$g$'?

Answer 1

No, consider the function $g(x) = x^{1/4}$ which when squared is concave.