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

$y = x^{4} - x^{3}$

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

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

Find $f (- x)$ .

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

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

Simplify each term.

Apply the product rule to $- x$ .

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

Raise $- 1$ to the power of $4$ .

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

Multiply $x^{4}$ by $1$ .

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

Apply the product rule to $- x$ .

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

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

Move ${(- 1)}^{3}$ .

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

Multiply ${(- 1)}^{3}$ by $- 1$ .

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $3$ and $1$ .

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

Raise $- 1$ to the power of $4$ .

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

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

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

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

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

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

Since $x^{4} + x^{3}$ $\neq$ $x^{4} - x^{3}$ , the function is not even.

The function is not even

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

Find $- f (x)$ .

Multiply $x^{4} - x^{3}$ by $- 1$ .

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

Apply the distributive property.

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

Since $x^{4} + x^{3}$ $\neq$ $- x^{4} + x^{3}$ , the function is not odd.

The function is not odd

The function is neither odd nor even