How to draw a $(f \circ g)(x)$ function without a rule and with graphs of $f(x)$ and $g(x)$?

Question

Is it possible to draw the composite function $(f \circ g)(x)$ without having a rule for the two functions $f$ and $g$ and only having graphs for these two functions?

For example these two functions

Answer 1

It is possible but a bit tedious. See below the gif showing how you may construct the composition $f\circ g$ point by point using the line $y=x$.

I chose the functions to look similar to those in your picture:

Answer 2

Let $x_0$ in the domain of $g$. You do have the point $(x_0, g (x_0))$. For $f (g (x_0))$ to make sense, $g (x_0)$ has to be in the domain of $f$. In this case we clearly can get the point $(g (x_0), f (g (x_0)))$ which is none other than $f\circ g (x_0)$.This way you can have the graph of $f\circ g$ point by point.