Precalculus Examples

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

Step 1

Move the negative in front of the fraction.

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

Step 2

Find $f (- x)$ .

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Step 2.1

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

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

Apply the product rule to $- x$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $- - x$ .

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Step 2.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.3.2

Multiply $x$ by $1$ .

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

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Factor $- 1$ out of $- x^{3}$ .

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

Step 2.3.2

Factor $- 1$ out of $x$ .

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

Step 2.3.3

Factor $- 1$ out of $- (x^{3}) - 1 (- x)$ .

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

Step 2.3.4

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

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

Step 2.3.5

Factor $- 1$ out of $- (x^{3} - x) - 1 (- 1)$ .

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

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Rewrite $- (x^{3} - x - 1)$ as $- 1 (x^{3} - x - 1)$ .

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

Step 2.3.6.2

Move the negative in front of the fraction.

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

Step 2.3.6.3

Multiply $- 1$ by $- 1$ .

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

Step 2.3.6.4

Multiply $\frac{4}{x^{3} - x - 1}$ by $1$ .

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

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 $\frac{4}{x^{3} - x - 1}$ $\neq$ $- \frac{4}{x^{3} - x + 1}$ , 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

Multiply $- (- \frac{4}{x^{3} - x + 1})$ .

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Step 4.1.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.2

Multiply $\frac{4}{x^{3} - x + 1}$ by $1$ .

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

Step 4.2

Since $\frac{4}{x^{3} - x - 1}$ $\neq$ $\frac{4}{x^{3} - x + 1}$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6