Precalculus Examples

$y = 8 x^{12} - 8 x^{4} + 6 x + 22$

Step 1

Write $y = 8 x^{12} - 8 x^{4} + 6 x + 22$ as a function.

$f (x) = 8 x^{12} - 8 x^{4} + 6 x + 22$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = 8 {(- x)}^{12} - 8 {(- x)}^{4} + 6 (- x) + 22$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = 8 ({(- 1)}^{12} x^{12}) - 8 {(- x)}^{4} + 6 (- x) + 22$

Step 2.2.2

Raise $- 1$ to the power of $12$ .

$f (- x) = 8 (1 x^{12}) - 8 {(- x)}^{4} + 6 (- x) + 22$

Step 2.2.3

Multiply $x^{12}$ by $1$ .

$f (- x) = 8 x^{12} - 8 {(- x)}^{4} + 6 (- x) + 22$

Step 2.2.4

Apply the product rule to $- x$ .

$f (- x) = 8 x^{12} - 8 ({(- 1)}^{4} x^{4}) + 6 (- x) + 22$

Step 2.2.5

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

$f (- x) = 8 x^{12} - 8 (1 x^{4}) + 6 (- x) + 22$

Step 2.2.6

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

$f (- x) = 8 x^{12} - 8 x^{4} + 6 (- x) + 22$

Step 2.2.7

Multiply $- 1$ by $6$ .

$f (- x) = 8 x^{12} - 8 x^{4} - 6 x + 22$

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 $8 x^{12} - 8 x^{4} - 6 x + 22$ $\neq$ $8 x^{12} - 8 x^{4} + 6 x + 22$ , the function is not even.

The function is not even

Step 4

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

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Step 4.1

Find $- f (x)$ .

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Step 4.1.1

Multiply $8 x^{12} - 8 x^{4} + 6 x + 22$ by $- 1$ .

$- f (x) = - (8 x^{12} - 8 x^{4} + 6 x + 22)$

Step 4.1.2

Apply the distributive property.

$- f (x) = - (8 x^{12}) - (- 8 x^{4}) - (6 x) - 1 \cdot 22$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Multiply $8$ by $- 1$ .

$- f (x) = - 8 x^{12} - (- 8 x^{4}) - (6 x) - 1 \cdot 22$

Step 4.1.3.2

Multiply $- 8$ by $- 1$ .

$- f (x) = - 8 x^{12} + 8 x^{4} - (6 x) - 1 \cdot 22$

Step 4.1.3.3

Multiply $6$ by $- 1$ .

$- f (x) = - 8 x^{12} + 8 x^{4} - 6 x - 1 \cdot 22$

Step 4.1.3.4

Multiply $- 1$ by $22$ .

$- f (x) = - 8 x^{12} + 8 x^{4} - 6 x - 22$

Step 4.2

Since $8 x^{12} - 8 x^{4} - 6 x + 22$ $\neq$ $- 8 x^{12} + 8 x^{4} - 6 x - 22$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6