Question

How do you determine if `f(x) = x^3 + x` is an even or odd function?

Answer 1

It is an odd function since `f(-x)=-f(x) AA x in RR`

Explanation:

A function `f(x)` is even if `f(-x)=f(x)AAx in RR`, and it is odd if `f(-x)=-f(x)AAx`.

So in this case hence it is odd.

For example, select `x=2`.
Then `f(-x)=f(-2)=(-2)^3-2=-10=-f(x)=2^2+2=10`.

Note also that all odd functions which are piecewise continuous and differentiable and having period `2L` may be represented as an infinite Fourier power series containing only sine terms, ie of the form
`sum_(n=1)^oo b_n sin((npix)/L)`, where `b_n=1/Lint_0^L f(x)sin((npix)/L)dx`

Answer 2

odd function

Explanation:

To determine if a function is even/odd the following applies.

• If f(x) = f( -x) then f(x) is even , `AAx`

Even functions have symmetry about the t-axis.

• If f(-x) = - f(x) then f(x) is odd , `AAx`

Odd functions have symmetry about the origin.

Test for even :

f( -x) = `(-x)^3 + (-x) = - x^3 - x ≠ f(x)` , hence not even

Test for odd :

`- f(x) = - (x^3 + x) = - x^3 - x =f(-x)`, hence function is odd.

Here is the graph of f(x). Note symmetry about O.
graph{x^3+x [-10, 10, -5, 5]}