Determine algebraically whether the function f(x)= x^3 -2x+1 is odd, even or neither.

Answer 1

Test it. To be even, f(x) must satisfy

f(-x) = f(x).

To be odd, f(x) must satisfy

f(-x) = -f(x)

Replace x with -x, compare f(x) with f(-x), and see that neither condition is met.

Answer 2

For an even function, f(-x) = f(x)

For an odd function, f(-x) = - f(x)

So we substitute in (-x) into the function: (-x)³ - 2(-x) + 1

And simplify: -x³ + 2x + 1

The is not the same as the original function, so it is not even.

If I factor out a negative, -(x³ - 2x - 1), the function inside the parentheses is not the same as the original function, so it is not odd.

Therefore it must be neither.