One-Dimensional Time-Independent Schrodinger

The one dimensional parity operator (pie) is defined by attached. in other wards (phi) changes x into -x everywhere in the function

(a) is it a Hermitian  operator?

(b) For what potentials V(x), is it possible to find a set of wave functions which are high eign- functions of the parity operator in solution of the one-dimensional time independent Schrodinger Eq?

I believe a) is yes, but am not sure 100%. I however do not now had to do part b). Detail solution please.