Precalculus Examples

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

Step 1

Find $f (- x)$ .

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Step 1.1

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

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

Apply the product rule to $- x$ .

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

Step 1.2.2

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

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

Step 1.3

Simplify with factoring out.

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Step 1.3.1

Move the negative in front of the fraction.

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

Step 1.3.2

Factor $- 1$ out of $- x$ .

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

Step 1.3.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 1.3.4

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

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

Step 1.3.5

Simplify the expression.

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Step 1.3.5.1

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

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

Step 1.3.5.2

Move the negative in front of the fraction.

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

Step 1.3.5.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.5.4

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

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

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

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

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

Step 3.2

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

The function is not odd

Step 4

The function is neither odd nor even

Step 5