Determine if Odd, Even, or Neither -x^5-1

$- x^{5} - 1$

Step 1

Write $- x^{5} - 1$ as a function.

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

Step 2

Find $f (- x)$ .

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

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

Simplify each term.

Apply the product rule to $- x$ .

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

Multiply $- 1$ by ${(- 1)}^{5}$ by adding the exponents.

Move ${(- 1)}^{5}$ .

$f (- x) = {(- 1)}^{5} \cdot (- 1 x^{5}) - 1$

Multiply ${(- 1)}^{5}$ by $- 1$ .

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

$f (- x) = {(- 1)}^{5} \cdot ((- 1) x^{5}) - 1$

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

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

Add $5$ and $1$ .

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

Raise $- 1$ to the power of $6$ .

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

Multiply $x^{5}$ by $1$ .

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

Step 3

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

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

$x^{5} - 1$ $\neq$ $- x^{5} - 1$

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

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

Apply the distributive property.

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

Since $x^{5} - 1$ $\neq$ $x^{5} + 1$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6