Determine if functions are odd, even or and their symmetries, if any

Question

a. f(x)=x^3+x

b. g(x)=Pi

c. h(x)= sin x / x^2 +5

d. k(x) = cos x / e^x

Answer 1

a) odd

b) even

c) as given neither, but sinx/(x^2+5) is even

d) neither

graph them and look for symmetry

all odd powers often odd

all even powers often even

Answer 2

A function, s(x), is

1. odd if s(–x) = –s(x)
2. even if s(–x) = s(x)
3. neither if s(–x) ≠ –s(x) and s(–x) ≠ s(x)

For example

a. f(x) = x³ + x and f(–x) = (–x)³ + (–x) = –(x³ + x) = –f(x)

so f(x) is an odd function

You can work through the answers for the given functions g(x), h(x), and k(x) in the same manner.

(Hint: Of these three remaining functions, one is even, one is odd, and one is neither.)

Hope this helps. Just add a comment if you have any questions.