Determine if Odd, Even, or Neither g(x)=x^6-2x

$g (x) = x^{6} - 2 x$

Step 1

Find $f (- x)$ .

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

$g (- x) = {(- x)}^{6} - 2 (- x)$

Simplify each term.

Apply the product rule to $- x$ .

$g (- x) = {(- 1)}^{6} x^{6} - 2 (- x)$

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

$g (- x) = 1 x^{6} - 2 (- x)$

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

$g (- x) = x^{6} - 2 (- x)$

Multiply $- 1$ by $- 2$ .

$g (- x) = x^{6} + 2 x$

Step 2

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

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

$x^{6} + 2 x$ $\neq$ $x^{6} - 2 x$

Since $x^{6} + 2 x$ $\neq$ $x^{6} - 2 x$ , the function is not even.

The function is not even

Step 3

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

Find $- f (x)$ .

Multiply $x^{6} - 2 x$ by $- 1$ .

$- f (x) = - (x^{6} - 2 x)$

Apply the distributive property.

$- f (x) = - x^{6} - (- 2 x)$

Multiply $- 2$ by $- 1$ .

$- f (x) = - x^{6} + 2 x$

Since $x^{6} + 2 x$ $\neq$ $- x^{6} + 2 x$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5