What function corresponds to the following graph description?

Question

This is almost like a puzzle.

This is the graph of $\csc(x).$

Now imagine I take each half-period and rotate it $180^{\circ},$ or flip it vertically, for both operations it would be with reference to the min/max point.

What kind of function would I need to get a graph like this?

Since my example graph is not precise, any function that creates something that looks similar will do.

Thanks for your consideration.

Answer 1

$$f(x) = -\csc(x) + 2 \sin (x)$$

Desmos plot:

Answer 2

Note, it might be possible to find more beautiful formula, nonetheless, here it is

$$\forall_{k \in \mathbb{Z}} \left\{ \begin{array}{ccc} -\csc{x} + 2& \textrm{for} & x \in (k\pi,(k + 1)\pi)\\ -\csc{x} - 2 & \textrm{for} & x\in ((k+1)\pi, (k+2)\pi) \end{array} \right. $$

Function prooposed by John Hughes is very closed to this formula, however, it is not exactly the same, and is not the exact representation of the transformation outlined in the question.

Therefore, depending on the use case you might prefer to choose one of two.