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

$y = x^{3} - x$

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

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

Find $f (- x)$ .

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

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

Simplify each term.

Apply the product rule to $- x$ .

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

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

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

Multiply $- (- x)$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

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

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

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

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

The function is not even

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

Find $- f (x)$ .

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

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

Apply the distributive property.

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

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

The function is odd