Background. I'm studying the graph of functions. I seem to understand shifting and scaling, but I've been trying to understand compositions in general --- and it is looking impossible to infer the graphical representation of two arbitrary functions composed. (I'm reading Calculus by James Stewart and the book seems to not expose any theory for the effect of compositions in graphs.)
Question. For example, although I can easily see the shape of $x^2$, $-x^2$, $\sqrt{x}$ and $1 - x^2$, it doesn't seem easy to infer the shape of $\sqrt{1 - x^2}$, which is the composition of $1 - x^2$ with $\sqrt{x}$. How could I infer the shape of $\sqrt{1 - x^2}$ from the shape of $\sqrt{x}$ and of $1 - x^2$?
Some comments. If I work hard enough, I can give an explanation as to why that produces the upper half-circle. But, of course, if I change the inner function and consider $\sqrt{\sin x}$, then I can't explain it at all in terms of $\sin x$ --- I've no idea. If I consider $\sqrt{e^x}$, I need to rewrite it to $e^{x/2}$ to explain its shape --- shifting the problem of a composition to a new problem, which is not really what I was hoping for. Is it the case that this is too difficult in general?