Determine if Odd, Even, or Neither -4x^3+4x

$- 4 x^{3} + 4 x$

Step 1

Write $- 4 x^{3} + 4 x$ as a function.

$f (x) = - 4 x^{3} + 4 x$

Step 2

Find $f (- x)$ .

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

$f (- x) = - 4 {(- x)}^{3} + 4 (- x)$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = - 4 ({(- 1)}^{3} x^{3}) + 4 (- x)$

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

$f (- x) = - 4 (- x^{3}) + 4 (- x)$

Multiply $- 1$ by $- 4$ .

$f (- x) = 4 x^{3} + 4 (- x)$

Multiply $- 1$ by $4$ .

$f (- x) = 4 x^{3} - 4 x$

Step 3

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

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

Since $4 x^{3} - 4 x$ $\neq$ $- 4 x^{3} + 4 x$ , 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 $- 4 x^{3} + 4 x$ by $- 1$ .

$- f (x) = - (- 4 x^{3} + 4 x)$

Apply the distributive property.

$- f (x) = - (- 4 x^{3}) - (4 x)$

Multiply.

Multiply $- 4$ by $- 1$ .

$- f (x) = 4 x^{3} - (4 x)$

Multiply $4$ by $- 1$ .

$- f (x) = 4 x^{3} - 4 x$

Since $4 x^{3} - 4 x = 4 x^{3} - 4 x$ , the function is odd.

The function is odd

Step 5