Precalculus Examples

$f (x) = \frac{1}{2} x^{3} - 2 x$

Step 1

Combine $\frac{1}{2}$ and $x^{3}$ .

$f (x) = \frac{x^{3}}{2} - 2 x$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = \frac{{(- x)}^{3}}{2} - 2 (- x)$

Step 2.2

Simplify each term.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Apply the product rule to $- x$ .

$f (- x) = \frac{{(- 1)}^{3} x^{3}}{2} - 2 (- x)$

Step 2.2.1.2

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

$f (- x) = \frac{- x^{3}}{2} - 2 (- x)$

Step 2.2.2

Move the negative in front of the fraction.

$f (- x) = - \frac{x^{3}}{2} - 2 (- x)$

Step 2.2.3

Multiply $- 1$ by $- 2$ .

$f (- x) = - \frac{x^{3}}{2} + 2 x$

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 $- \frac{x^{3}}{2} + 2 x$ $\neq$ $\frac{x^{3}}{2} - 2 x$ , 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 $\frac{x^{3}}{2} - 2 x$ by $- 1$ .

$- f (x) = - (\frac{x^{3}}{2} - 2 x)$

Step 4.1.2

Apply the distributive property.

$- f (x) = - \frac{x^{3}}{2} - (- 2 x)$

Step 4.1.3

Multiply $- 2$ by $- 1$ .

$- f (x) = - \frac{x^{3}}{2} + 2 x$

Step 4.2

Since $- \frac{x^{3}}{2} + 2 x = - \frac{x^{3}}{2} + 2 x$ , the function is odd.

The function is odd

Step 5