Determine if Odd, Even, or Neither x^6+x

$x^{6} + x$

Write $x^{6} + x$ as a function.

$f (x) = x^{6} + x$

Find $f (- x)$ .

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = {(- x)}^{6} - x$

Remove parentheses.

$f (- x) = {(- x)}^{6} - x$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{6} x^{6} - x$

Raise $- 1$ to the power of $6$ .

$f (- x) = 1 x^{6} - x$

Multiply $x^{6}$ by $1$ .

$f (- x) = x^{6} - x$

A function is even if $f (- x) = f (x)$ .

Check if $f (- x) = f (x)$ .

$x^{6} - x$ $\neq$ $x^{6} + x$

Since $x^{6} - x$ $\neq$ $x^{6} + x$ , the function is not even.

The function is not even

A function is odd if $f (- x) = - f (x)$ .

Find $- f (x)$ .

Multiply $x^{6} + x$ by $- 1$ .

$- f (x) = - (x^{6} + x)$

Apply the distributive property.

$- f (x) = - x^{6} - x$

Since $x^{6} - x$ $\neq$ $- x^{6} - x$ , the function is not odd.

The function is not odd

The function is neither odd nor even