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?
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Welcome to MSE. Please read this text about how to ask a good question. – José Carlos Santos Jun 19 '22 at 12:38
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The question of "what does $f(g(x))$ look like in general" is rather broad, although there are some facts you can use, like if both $f$ and $g$ are increasing at a point then so is $f(g(x)).$ Similar with decreasing. – Adam Rubinson Jun 19 '22 at 13:26
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There is no general answer to this question. The function $g$ manipulates the domain of $f$ in a way that can be hard to describe explicitly. You can begin to see some hints at https://math.stackexchange.com/questions/3307507/mechanics-of-horizontal-stretching-and-shrinking/3307525#3307525 – Ethan Bolker Jun 19 '22 at 15:15
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You could plot individual points with (a variation of) the web diagram. Given input value $x$, move vertically to $(x,f(x))$ on the graph of $f$, from there move horizontally to the line $y=x$, from there vertically to $(f(x),g(f(x)))$ on the graph of $g$, and from there horizontally to $(x,g(f(x)))$. Not useful for the big picture but ok for a few points. – Ned Jun 19 '22 at 17:29
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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.
Artur Wiadrowski
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