Algebra Examples

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

Find $f (- x)$ .

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

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

Apply the product rule to $- x$ .

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

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Move ${(- 1)}^{3}$ .

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

Multiply ${(- 1)}^{3}$ by $- 1$ .

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $3$ and $1$ .

$f (- x) = {(- 1)}^{4} (\frac{1}{2}) \cdot x^{3}$

Simplify the expression.

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

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

Multiply $\frac{1}{2}$ by $1$ .

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

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

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

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

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

$\frac{x^{3}}{2}$ $\neq$ $- \frac{1}{2} x^{3}$

Since $\frac{x^{3}}{2}$ $\neq$ $- \frac{1}{2} x^{3}$ , the function is not even.

The function is not even

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

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Find $- f (x)$ .

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

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

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

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

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

The function is odd