If we know the graph of $f(x)$ and of $g(x)$, is there a way to graph their composition $f(g(x))$?

Question

My question is that if we know the graph of $f(x)$ and of $g(x)$, s there a way to graph $f(g(x))$

Example: $\sin (\ln (x))$

How do we reach this graph? How does this graph relate to its parent functions?

Answer 1

Notice that if $g(x)=x^{2}$, then for $x < 1$ the graph of $f(g(x))$ will be squished compared to $f(x)$. This is only an example.

Take a point on the $x$ axis and apply to its value $g(x)$. It will be moved somewhere.

Now, drawing the graph of $f(g(x))$ is like constructing a $2$-dimensional graph with the $x$ axis replaced with the second axis from above. The value corresponding to that $2$ of the second axis should be $f(2)$.

This can of course be applied generally. However, if you take $g(x)=\sin x$ or consider in my example also negative values of $x$ you will see that in your new graph you will have multiple $x$ values corresponding to a single $y$ which is not allowed.