Precalculus Examples

$g (x) = x^{4} - x$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$g (- x) = {(- x)}^{4} - (- x)$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$g (- x) = {(- 1)}^{4} x^{4} - (- x)$

Step 1.2.2

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

$g (- x) = 1 x^{4} - (- x)$

Step 1.2.3

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

$g (- x) = x^{4} - (- x)$

Step 1.2.4

Multiply $- (- x)$ .

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Step 1.2.4.1

Multiply $- 1$ by $- 1$ .

$g (- x) = x^{4} + 1 x$

Step 1.2.4.2

Multiply $x$ by $1$ .

$g (- x) = x^{4} + x$

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 $x^{4} + x$ $\neq$ $x^{4} - x$ , 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 $x^{4} - x$ by $- 1$ .

$- f (x) = - (x^{4} - x)$

Step 3.1.2

Apply the distributive property.

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

Step 3.2

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

The function is not odd

Step 4

The function is neither odd nor even

Step 5