If f(x) = f(-x) for all x, then the graph of f(x) is symmetric with respect to the y-axis, and the function f(x) is called an even function. If f(x) = -f(-x) for all x, the graph of f(x) is anti-symmetric with respect to the origin, and we call such a function an odd function

a. Show that any function can be written as the sum of an odd function plus an even function. List as many even and odd functions as you can.

b. State what conditions must be true for a polynomial to be even, or to be odd.

\c. Show that the product of two even functions is even; the product of two odd functions is even; and the product of an odd and even function is odd.

d. Replace in c above the word product by either quotient or power and deduce the parity of the resulting function.

e. Deduce from the above results that the sign/parity of a function follows algebraic rules.

f. Find the even and odd parts of the following functions:

(i) f(x) = x7 + 3x4 + 6x + 2

(ii) f(x) = (sin(x) + 3) sinh2(x) exp(-x2)