# Is the function f(x)=(1/x^3+x)^5 even, odd or neither?

Nov 17, 2015

$f \left(x\right) = {\left(\frac{1}{x} ^ 3 + x\right)}^{5}$ is odd.

#### Explanation:

A function $f \left(x\right)$ is even if and only if $f \left(- x\right) = f \left(x\right)$.
A function $f \left(x\right)$ is odd if and only if $f \left(- x\right) = - f \left(x\right)$.

To check this function then, we will look at $f \left(- x\right)$.

$f \left(- x\right) = {\left(\frac{1}{- x} ^ 3 + \left(- x\right)\right)}^{5} = {\left(- \frac{1}{x} ^ 3 - x\right)}^{5}$

$\implies f \left(- x\right) = {\left(\left(- 1\right) \left(\frac{1}{x} ^ 3 + x\right)\right)}^{5} = {\left(- 1\right)}^{5} {\left(\frac{1}{x} ^ 3 + x\right)}^{5}$

$\implies f \left(- x\right) = - {\left(\frac{1}{x} ^ 3 + x\right)}^{5} = - f \left(x\right)$

Thus in this case $f \left(x\right)$ is odd.