Precalculus Examples

$f (x) = x^{3} - x$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = {(- x)}^{3} - (- x)$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{3} x^{3} - (- x)$

Step 1.2.2

Raise $- 1$ to the power of $3$ .

$f (- x) = - x^{3} - (- x)$

Step 1.2.3

Multiply $- (- x)$ .

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Step 1.2.3.1

Multiply $- 1$ by $- 1$ .

$f (- x) = - x^{3} + 1 x$

Step 1.2.3.2

Multiply $x$ by $1$ .

$f (- x) = - x^{3} + x$

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 $- x^{3} + x$ $\neq$ $x^{3} - x$ , 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 $x^{3} - x$ by $- 1$ .

$- f (x) = - (x^{3} - x)$

Step 3.1.2

Apply the distributive property.

$- f (x) = - x^{3} + x$

Step 3.2

Since $- x^{3} + x = - x^{3} + x$ , the function is odd.

The function is odd

Step 4