Precalculus Examples

$h (x) = x^{2} - x^{8}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$h (- x) = {(- x)}^{2} - {(- x)}^{8}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$h (- x) = {(- 1)}^{2} x^{2} - {(- x)}^{8}$

Step 1.2.2

Raise $- 1$ to the power of $2$ .

$h (- x) = 1 x^{2} - {(- x)}^{8}$

Step 1.2.3

Multiply $x^{2}$ by $1$ .

$h (- x) = x^{2} - {(- x)}^{8}$

Step 1.2.4

Apply the product rule to $- x$ .

$h (- x) = x^{2} - ({(- 1)}^{8} x^{8})$

Step 1.2.5

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

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Step 1.2.5.1

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

$h (- x) = x^{2} + {(- 1)}^{8} \cdot (- 1 x^{8})$

Step 1.2.5.2

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

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Step 1.2.5.2.1

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

$h (- x) = x^{2} + {(- 1)}^{8} \cdot ((- 1) x^{8})$

Step 1.2.5.2.2

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

$h (- x) = x^{2} + {(- 1)}^{8 + 1} x^{8}$

Step 1.2.5.3

Add $8$ and $1$ .

$h (- x) = x^{2} + {(- 1)}^{9} x^{8}$

Step 1.2.6

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

$h (- x) = x^{2} - x^{8}$

Step 2

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

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Step 2.1

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

Step 2.2

Since $x^{2} - x^{8} = x^{2} - x^{8}$ , the function is even.

The function is even

Step 3