Answer
See graph
Work Step by Step
$$\eqalign{
& f\left( x \right) = \tan \left( {\sin \pi x} \right) \cr
& \cr
& \left( {\text{a}} \right){\text{Graph of the function using a CAS }}\left( {{\text{shown below}}} \right) \cr
& \cr
& \left( {\text{b}} \right){\text{From the graph, we can notice that the function is odd,}} \cr
& {\text{so it is symmetric with respect to the origin}}{\text{.}} \cr
& {\text{analitically}} \cr
& f\left( { - x} \right) = \tan \left( {\sin \left( { - \pi x} \right)} \right) \cr
& f\left( { - x} \right) = \tan \left( { - \sin \left( {\pi x} \right)} \right) \cr
& f\left( { - x} \right) = - \tan \left( {\sin \left( {\pi x} \right)} \right) \cr
& f\left( { - x} \right) = - f\left( x \right),{\text{ odd function}}{\text{.}} \cr
& \cr
& \left( {\text{c}} \right){\text{Yes, the function is periodic, The period is }}T = 2. \cr
& \cr
& \left( {\text{d}} \right){\text{The extrema on }}\left( { - 1,1} \right){\text{ are:}} \cr
& {\text{Relative minimum at the point }}\left( { - 0.5, - 1.56} \right) \cr
& {\text{Relative maximum at the point }}\left( {0.5,1.56} \right) \cr
& \cr
& \left( {\text{e}} \right){\text{From the graph we can notice that the graph is }} \cr
& {\text{concave down on the interval }}\left( {0,1} \right) \cr} $$
