We’re trying to find a set of three functions that satisfied a certain property. Look closely at it, and you see that what it really means is that, firstly, all three lines are concurrent at exactly three points. Then, their derivatives have the same property. This is recursive. Consider that the derivative of a sine wave is also a sine wave. We then have a relatively simple solution to the problem, but which is tricky to figure out.
\left. \begin{array}{l} \text{if $0\le x\lt3\pi$:} & \sin x \\ \text{if $x\ge3\pi$:} & f(-x) \\ \text{if $x\lt0$:} & e^x \\ \end{array} \right\} =f(x) \\[8pt] g(x) = -f(x)\\[8pt] h(x) = 0\\[16pt] f\circledcirc g\circledcirc hSolution to Fascinating Frequencies BONUS
\text{let }x,\,y,\,z\in\R,\;x\leq y\leq z\\[8pt]
\text{if }\;\exists!\;\{x,\,y,\,z\}:
\operatorname fa=\operatorname gb=\operatorname hc\;\;
\forall\;\;\{a,\,b,\,c\}\equiv\{x,\,y,\,z\},\\[4pt]
\text{then }\;f\bigcirc g\bigcirc h\\[8pt]
\text{if }\ f\bigcirc g\bigcirc h\;\ \text{and }\ f’\circledcirc g’\circledcirc h’,\ \;\text{then}\;f\circledcirc g\circledcirc h