Determine if Odd, Even, or Neither f(x)=x^11+x^5

$f (x) = x^{11} + x^{5}$

Step 1

Find $f (- x)$ .

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

$f (- x) = {(- x)}^{11} + {(- x)}^{5}$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{11} x^{11} + {(- x)}^{5}$

Raise $- 1$ to the power of $11$ .

$f (- x) = - x^{11} + {(- x)}^{5}$

Apply the product rule to $- x$ .

$f (- x) = - x^{11} + {(- 1)}^{5} x^{5}$

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

$f (- x) = - x^{11} - x^{5}$

Step 2

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

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

Since $- x^{11} - x^{5}$ $\neq$ $x^{11} + x^{5}$ , 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 $x^{11} + x^{5}$ by $- 1$ .

$- f (x) = - (x^{11} + x^{5})$

Apply the distributive property.

$- f (x) = - x^{11} - x^{5}$

Since $- x^{11} - x^{5} = - x^{11} - x^{5}$ , the function is odd.

The function is odd

Step 4