Determine if Odd, Even, or Neither g(x)=x^12+x^5

$g (x) = x^{12} + x^{5}$

Find $f (- x)$ .

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

$g (- x) = {(- x)}^{12} + {(- x)}^{5}$

Simplify each term.

Apply the product rule to $- x$ .

$g (- x) = {(- 1)}^{12} x^{12} + {(- x)}^{5}$

Raise $- 1$ to the power of $12$ .

$g (- x) = 1 x^{12} + {(- x)}^{5}$

Multiply $x^{12}$ by $1$ .

$g (- x) = x^{12} + {(- x)}^{5}$

Apply the product rule to $- x$ .

$g (- x) = x^{12} + {(- 1)}^{5} x^{5}$

Raise $- 1$ to the power of $5$ .

$g (- x) = x^{12} - x^{5}$

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

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

$x^{12} - x^{5}$ $\neq$ $x^{12} + x^{5}$

Since $x^{12} - x^{5}$ $\neq$ $x^{12} + x^{5}$ , the function is not even.

The function is not even

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

Find $- f (x)$ .

Multiply $x^{12} + x^{5}$ by $- 1$ .

$- f (x) = - (x^{12} + x^{5})$

Apply the distributive property.

$- f (x) = - x^{12} - x^{5}$

Since $x^{12} - x^{5}$ $\neq$ $- x^{12} - x^{5}$ , the function is not odd.

The function is not odd

The function is neither odd nor even