Problem Teachers' Notes Hint Solution

Operating Machines

For the first case we get the set $x^n$ for any integer $n$.

For the second case we get $\frac{x^n}{n!}$ for any non-negative integer $n$

The 'fixed points' are $1$ and $e^x$

Being more clever, there might be possibilities involving non-integrable and non-differentiable functions for the second part