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

$x^{3} - 3 x$

Step 1

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

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

Step 2

Find $f (- x)$ .

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

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

Simplify each term.

Apply the product rule to $- x$ .

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

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

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

Multiply $- 1$ by $- 3$ .

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

Step 3

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

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

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

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

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

Apply the distributive property.

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

Multiply $- 3$ by $- 1$ .

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

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

The function is odd

Step 5