Precalculus Examples

$- \frac{2}{3} x^{3}$

Step 1

Write $- \frac{2}{3} x^{3}$ as a function.

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Move $2$ to the left of $x^{3}$ .

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

Step 3

Find $f (- x)$ .

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Step 3.1

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

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Apply the product rule to $- x$ .

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

Step 3.2.2

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

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

Step 3.2.3

Combine exponents.

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Step 3.2.3.1

Factor out negative.

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

Step 3.2.3.2

Multiply $2$ by $- 1$ .

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

Step 3.3

Move the negative in front of the fraction.

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

Step 4

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

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Step 4.1

Check if $f (- x) = f (x)$ .

Step 4.2

Since $\frac{2 x^{3}}{3}$ $\neq$ $- \frac{2 x^{3}}{3}$ , the function is not even.

The function is not even

Step 5

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

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Step 5.1

Multiply $- (- \frac{2 x^{3}}{3})$ .

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Step 5.1.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.2

Multiply $\frac{2 x^{3}}{3}$ by $1$ .

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

Step 5.2

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

The function is odd

Step 6