Precalculus Examples

$f (x) = 5 x^{6} - 3 x^{4} + 3 x^{2} - x$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = 5 {(- x)}^{6} - 3 {(- x)}^{4} + 3 {(- x)}^{2} - (- x)$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = 5 ({(- 1)}^{6} x^{6}) - 3 {(- x)}^{4} + 3 {(- x)}^{2} - (- x)$

Step 1.2.2

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

$f (- x) = 5 (1 x^{6}) - 3 {(- x)}^{4} + 3 {(- x)}^{2} - (- x)$

Step 1.2.3

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

$f (- x) = 5 x^{6} - 3 {(- x)}^{4} + 3 {(- x)}^{2} - (- x)$

Step 1.2.4

Apply the product rule to $- x$ .

$f (- x) = 5 x^{6} - 3 ({(- 1)}^{4} x^{4}) + 3 {(- x)}^{2} - (- x)$

Step 1.2.5

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

$f (- x) = 5 x^{6} - 3 (1 x^{4}) + 3 {(- x)}^{2} - (- x)$

Step 1.2.6

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

$f (- x) = 5 x^{6} - 3 x^{4} + 3 {(- x)}^{2} - (- x)$

Step 1.2.7

Apply the product rule to $- x$ .

$f (- x) = 5 x^{6} - 3 x^{4} + 3 ({(- 1)}^{2} x^{2}) - (- x)$

Step 1.2.8

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

$f (- x) = 5 x^{6} - 3 x^{4} + 3 (1 x^{2}) - (- x)$

Step 1.2.9

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

$f (- x) = 5 x^{6} - 3 x^{4} + 3 x^{2} - (- x)$

Step 1.2.10

Multiply $- (- x)$ .

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Step 1.2.10.1

Multiply $- 1$ by $- 1$ .

$f (- x) = 5 x^{6} - 3 x^{4} + 3 x^{2} + 1 x$

Step 1.2.10.2

Multiply $x$ by $1$ .

$f (- x) = 5 x^{6} - 3 x^{4} + 3 x^{2} + x$

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 $5 x^{6} - 3 x^{4} + 3 x^{2} + x$ $\neq$ $5 x^{6} - 3 x^{4} + 3 x^{2} - x$ , the function is not even.

The function is not even

Step 3

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

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Step 3.1

Find $- f (x)$ .

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Step 3.1.1

Multiply $5 x^{6} - 3 x^{4} + 3 x^{2} - x$ by $- 1$ .

$- f (x) = - (5 x^{6} - 3 x^{4} + 3 x^{2} - x)$

Step 3.1.2

Apply the distributive property.

$- f (x) = - (5 x^{6}) - (- 3 x^{4}) - (3 x^{2}) + x$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $5$ by $- 1$ .

$- f (x) = - 5 x^{6} - (- 3 x^{4}) - (3 x^{2}) + x$

Step 3.1.3.2

Multiply $- 3$ by $- 1$ .

$- f (x) = - 5 x^{6} + 3 x^{4} - (3 x^{2}) + x$

Step 3.1.3.3

Multiply $3$ by $- 1$ .

$- f (x) = - 5 x^{6} + 3 x^{4} - 3 x^{2} + x$

Step 3.1.3.4

Multiply $- - x$ .

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Step 3.1.3.4.1

Multiply $- 1$ by $- 1$ .

$- f (x) = - 5 x^{6} + 3 x^{4} - 3 x^{2} + 1 x$

Step 3.1.3.4.2

Multiply $x$ by $1$ .

$- f (x) = - 5 x^{6} + 3 x^{4} - 3 x^{2} + x$

Step 3.2

Since $5 x^{6} - 3 x^{4} + 3 x^{2} + x$ $\neq$ $- 5 x^{6} + 3 x^{4} - 3 x^{2} + x$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5