How to graph $y=f(x^2)=\sin(x^2)$? How to graph $y=f(x^2)=\sin(x^2)$?
I have substituted as follows:
$$\begin{cases} y=f(a)=\sin a\\ a=x^2\end{cases}.$$
Then if I graph this with the coordinate axes $y$ and $a$ I get the ordinary sine function. But this doesn't solve my problem. Is it possible to graph my example $f(x^2)$ with the axes $y$ and $x$?
 A: If you have the graphs of $y=f(x)$ and $y=g(x)$, you can create the graph of $y=g(f(x))$ from them easily in the following manner.
First, draw the graphs of $y=f(x)$ and $y=g(x)$ on the same set of axes, and additionally draw the line $y=x$ there as well.
To plot the point $(x,g(f(x)))$, start at the point $(x,0)$ on the horizontal axis. Then move vertically to the graph of $f$. You are now at the point $(x,f(x))$.
Move horizontally to the line $y=x$. You are now at the point $(f(x),f(x))$.
Move vertically to the graph of $g$. You are now at the point $(f(x),g(f(x)))$.
Move horizontally until you are directly over the starting point. You are now at the desired point $(x,g(f(x)))$ on the graph of $g\circ f$.
This is a standard trick that lets you evaluate any number of functions successively. If you need to find $f_n(\cdots f_3(f_2(f_1(x)))\cdots)$, start at $(x,0)$, then move up to $f_1$, over to the diagonal, up to $f_2$, over to the diagonal, up to $f_3$, over to the diagonal, etc. . It is especially useful when iterating a single function over and over.
