Determine if Odd, Even, or Neither y=x^9+x^3

$y = x^{9} + x^{3}$

Step 1

Write $y = x^{9} + x^{3}$ as a function.

$f (x) = x^{9} + x^{3}$

Step 2

Find $f (- x)$ .

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

$f (- x) = {(- x)}^{9} + {(- x)}^{3}$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{9} x^{9} + {(- x)}^{3}$

Raise $- 1$ to the power of $9$ .

$f (- x) = - x^{9} + {(- x)}^{3}$

Apply the product rule to $- x$ .

$f (- x) = - x^{9} + {(- 1)}^{3} x^{3}$

Raise $- 1$ to the power of $3$ .

$f (- x) = - x^{9} - x^{3}$

Step 3

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

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

$- x^{9} - x^{3}$ $\neq$ $x^{9} + x^{3}$

Since $- x^{9} - x^{3}$ $\neq$ $x^{9} + x^{3}$ , the function is not even.

The function is not even

Step 4

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

Find $- f (x)$ .

Multiply $x^{9} + x^{3}$ by $- 1$ .

$- f (x) = - (x^{9} + x^{3})$

Apply the distributive property.

$- f (x) = - x^{9} - x^{3}$

Since $- x^{9} - x^{3} = - x^{9} - x^{3}$ , the function is odd.

The function is odd

Step 5