Determine if Odd, Even, or Neither x^6-x^8

$x^{6} - x^{8}$

Write $x^{6} - x^{8}$ as a function.

$f (x) = x^{6} - x^{8}$

Find $f (- x)$ .

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

$f (- x) = {(- x)}^{6} - {(- x)}^{8}$

Simplify each term.

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Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{6} x^{6} - {(- x)}^{8}$

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

$f (- x) = 1 x^{6} - {(- x)}^{8}$

Multiply $x^{6}$ by $1$ .

$f (- x) = x^{6} - {(- x)}^{8}$

Apply the product rule to $- x$ .

$f (- x) = x^{6} - ({(- 1)}^{8} x^{8})$

Multiply $- 1$ by ${(- 1)}^{8}$ by adding the exponents.

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Move ${(- 1)}^{8}$ .

$f (- x) = x^{6} + {(- 1)}^{8} \cdot (- 1 x^{8})$

Multiply ${(- 1)}^{8}$ by $- 1$ .

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

$f (- x) = x^{6} + {(- 1)}^{8} \cdot ((- 1) x^{8})$

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

$f (- x) = x^{6} + {(- 1)}^{8 + 1} x^{8}$

Add $8$ and $1$ .

$f (- x) = x^{6} + {(- 1)}^{9} x^{8}$

Raise $- 1$ to the power of $9$ .

$f (- x) = x^{6} - x^{8}$

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

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

$x^{6} - x^{8} = x^{6} - x^{8}$

Since $x^{6} - x^{8} = x^{6} - x^{8}$ , the function is even.

The function is even

Determine if Odd, Even, or Neither x^6-x^8

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