A function f defined for -a < x < a is even if f(-x) = f(x) and is odd if f(-x) = -f(x) when -a < x < a. In this task we assume f is defined on such an interval, which might be the full real line (i.e. a = infinity).
a. Show that f(x) = x^2 is even and g(x) = x^3 is odd.
b. Write f(x) = 3x^3 +2x^2 - 5x +7 as a sum f(x) = e(x) + o(x), where e is even and o is odd.
c. Do the same for the function f(x) = 1/(1-x) on the domain -1 < x < 1. [Hint : multiply the numerator and denominator by 1+x].
d. Parts (b) and (c) suggest that it might always be possible to write f(x) = e(x) + o(x) where e is even and o is odd. Suppose that this is so, and use the definition of even and odd to write an equation expressing f(-x) in terms of e(x) and o(x).
e. You now have two equations: f(x) = e(x) + o(x) and the other one you obtained in part (d). Solve this system of equations for e(x) and o(x), and show that the resulting e(x) is even and the resulting o(x) is odd.
f. Based on your work in part (e), is it true or is it false that every function defined on the interval -a < x < a can be expressed as a sum of an even function and an odd function? Why?