# Is the function f(x)= -4x^2 + 4x even, odd or neither?

Oct 18, 2015

Neither

#### Explanation:

The quick way to spot whether a polynomial function in $x$ is odd or even is the powers of $x$ that occur. If they are all odd then the polynomial is odd. If they are all even then the polynomial is even. Note that a constant is an even power of $x$ - namely ${x}^{0}$.

By definition:

$f \left(x\right)$ is odd if $f \left(- x\right) = - f \left(x\right)$ for all $x$ in the domain.

$f \left(x\right)$ is even if $f \left(- x\right) = f \left(x\right)$ for all $x$ in the domain.

In our case, we find:

$f \left(- 1\right) = - 4 - 4 = - 8$

$f \left(1\right) = - 4 + 4 = 0$

So neither condition holds.

Given any function $f \left(x\right)$, it can be expressed uniquely as the sum of an even function and an odd function, defined as follows:

${f}_{e} \left(x\right) = \frac{f \left(x\right) + f \left(- x\right)}{2}$

${f}_{o} \left(x\right) = \frac{f \left(x\right) - f \left(- x\right)}{2}$

In our case we find ${f}_{e} \left(x\right) = - 4 {x}^{2}$ and ${f}_{o} \left(x\right) = 4 x$