Precalculus Examples

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

Step 1

Simplify.

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Step 1.1

Rewrite ${(x^{2} - 8)}^{2}$ as $(x^{2} - 8) (x^{2} - 8)$ .

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

Step 1.2

Expand $(x^{2} - 8) (x^{2} - 8)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

Add $2$ and $2$ .

$f (x) = x^{4} + x^{2} \cdot - 8 - 8 x^{2} - 8 \cdot - 8$

Step 1.3.1.2

Move $- 8$ to the left of $x^{2}$ .

$f (x) = x^{4} - 8 \cdot x^{2} - 8 x^{2} - 8 \cdot - 8$

Step 1.3.1.3

Multiply $- 8$ by $- 8$ .

$f (x) = x^{4} - 8 x^{2} - 8 x^{2} + 64$

Step 1.3.2

Subtract $8 x^{2}$ from $- 8 x^{2}$ .

$f (x) = x^{4} - 16 x^{2} + 64$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = {(- x)}^{4} - 16 {(- x)}^{2} + 64$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{4} x^{4} - 16 {(- x)}^{2} + 64$

Step 2.2.2

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

$f (- x) = 1 x^{4} - 16 {(- x)}^{2} + 64$

Step 2.2.3

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

$f (- x) = x^{4} - 16 {(- x)}^{2} + 64$

Step 2.2.4

Apply the product rule to $- x$ .

$f (- x) = x^{4} - 16 ({(- 1)}^{2} x^{2}) + 64$

Step 2.2.5

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

$f (- x) = x^{4} - 16 (1 x^{2}) + 64$

Step 2.2.6

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

$f (- x) = x^{4} - 16 x^{2} + 64$

Step 3

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

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Step 3.1

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

Step 3.2

Since $x^{4} - 16 x^{2} + 64 = x^{4} - 16 x^{2} + 64$ , the function is even.

The function is even

Step 4