Question

Suppose f ,g : R → R. Show that:

(a) If f is even then so is −f . If f is odd then so is −f .

(b) If f and g are both odd then f + g is odd, f g is even and f ◦ g is odd

(c) If f and g are both even then f + g, f g and f ◦ g are all even.

(d) If f is even and g is odd then f g is odd while f ◦ g and g ◦ f are both even.

(e) If f is both even and odd then f is identically zero (i.e. f (x) = 0 for all x ∈ R).