Task
For each function in this task, assume the domain is the largest set of real numbers for which the function value is a real number.
Let $f$ be the function defined by $f(x)=x^2$. Let $g$ be the function defined by $g(x)=\sqrt{x}$.

Sketch the graph of $y=f(g(x))$ and explain your reasoning.
 Sketch the graph of $y=g(f(x))$ and explain your reasoning.
IM Commentary
This task addresses an important issue about inverse functions. In this
case the function $f$ is the inverse of the function $g$ but $g$ is not
the inverse of $f$ unless the domain of $f$ is restricted.
This task includes an experimental GeoGebra worksheet, with the intent
that instructors might use it to more interactively demonstrate the
relevant content material. The file should be considered a draft
version, and feedback on it in the comment section is highly
encouraged, both in terms of suggestions for improvement and for ideas
on using it effectively. The file can be run via the free online
application GeoGebra, or run
locally if GeoGebra has been installed on a computer.