Determine if Odd, Even, or Neither f(x)=x^5+6x

$f (x) = x^{5} + 6 x$

Step 1

Find $f (- x)$ .

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

$f (- x) = {(- x)}^{5} + 6 (- x)$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{5} x^{5} + 6 (- x)$

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

$f (- x) = - x^{5} + 6 (- x)$

Multiply $- 1$ by $6$ .

$f (- x) = - x^{5} - 6 x$

Step 2

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

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

Since $- x^{5} - 6 x$ $\neq$ $x^{5} + 6 x$ , 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 $x^{5} + 6 x$ by $- 1$ .

$- f (x) = - (x^{5} + 6 x)$

Apply the distributive property.

$- f (x) = - x^{5} - (6 x)$

Multiply $6$ by $- 1$ .

$- f (x) = - x^{5} - 6 x$

Since $- x^{5} - 6 x = - x^{5} - 6 x$ , the function is odd.

The function is odd

Step 4