Question

Determine algebraically whether f(x) = x2(x2 + 9)(x3 + 2x) is even or odd.

Answer 1

`f(x) = x^(2)\cdot (x^(2)+9)\cdot (x^(3)+2\cdot x)` is an odd function.

Explanation:

Let be
`f(x) = x^(2)\cdot (x^(2)+9)\cdot (x^(3)+2\cdot x)`, by Algebra this expression can be rewritten as:

`f(x) = x^(3)\cdot (x^(2)+9)\cdot (x^(2)+2)`

Where
`x^(2) + 9` and
`x^(2)+ 2` are even functions, because they satisfy the condition that
`g(x) = g(-x)`, whereas
`x^(3)` is an odd function, as the condition of
`h(-x) = - h(x)` is observed. Then, the overall function is odd.