## Shape of a function (without convexity)

Investigate the shape of the following functions including their asymptotes and limits at boundary points of the domain of definition (you need not investigate convexity). Sketch their graphs.

• #### Variant 1

$$\displaystyle f(x)=\sin(\sin(x))$$

• #### Variant 2

$$\displaystyle f(x)=\sin(\pi \sin x)$$

• #### Variant 3

$$\displaystyle f(x)=\frac{x}{\sin x}$$ (in this variant you will probably not determine the extrema precisely, but try to at least estimate their positions)

• #### Variant 4

$$\displaystyle f(x)=2x^2(x - \sqrt{x^2 - 1})$$

• #### Variant 5

$$\displaystyle f(x)=\sin \left( \frac{1}{|x| + 2/(3 \pi)}\right)$$