Precalculus Examples

$f (x) = \frac{2 x^{3}}{3 x^{7} - 2}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = \frac{2 {(- x)}^{3}}{3 {(- x)}^{7} - 2}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = \frac{2 {(- 1)}^{3} x^{3}}{3 {(- x)}^{7} - 2}$

Step 1.2.2

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

$f (- x) = \frac{2 \cdot (- 1 x^{3})}{3 {(- x)}^{7} - 2}$

Step 1.2.3

Combine exponents.

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Step 1.2.3.1

Factor out negative.

$f (- x) = \frac{- (2 x^{3})}{3 {(- x)}^{7} - 2}$

Step 1.2.3.2

Multiply $2$ by $- 1$ .

$f (- x) = \frac{- 2 x^{3}}{3 {(- x)}^{7} - 2}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Apply the product rule to $- x$ .

$f (- x) = \frac{- 2 x^{3}}{3 ({(- 1)}^{7} x^{7}) - 2}$

Step 1.3.2

Raise $- 1$ to the power of $7$ .

$f (- x) = \frac{- 2 x^{3}}{3 (- x^{7}) - 2}$

Step 1.3.3

Multiply $- 1$ by $3$ .

$f (- x) = \frac{- 2 x^{3}}{- 3 x^{7} - 2}$

Step 1.4

Simplify with factoring out.

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Step 1.4.1

Move the negative in front of the fraction.

$f (- x) = - \frac{2 x^{3}}{- 3 x^{7} - 2}$

Step 1.4.2

Factor $- 1$ out of $- 3 x^{7}$ .

$f (- x) = - \frac{2 x^{3}}{- (3 x^{7}) - 2}$

Step 1.4.3

Rewrite $- 2$ as $- 1 (2)$ .

$f (- x) = - \frac{2 x^{3}}{- (3 x^{7}) - 1 \cdot 2}$

Step 1.4.4

Factor $- 1$ out of $- (3 x^{7}) - 1 (2)$ .

$f (- x) = - \frac{2 x^{3}}{- (3 x^{7} + 2)}$

Step 1.4.5

Simplify the expression.

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Step 1.4.5.1

Rewrite $- (3 x^{7} + 2)$ as $- 1 (3 x^{7} + 2)$ .

$f (- x) = - \frac{2 x^{3}}{- 1 (3 x^{7} + 2)}$

Step 1.4.5.2

Move the negative in front of the fraction.

$f (- x) = \frac{2 x^{3}}{3 x^{7} + 2}$

Step 1.4.5.3

Multiply $- 1$ by $- 1$ .

$f (- x) = 1 (\frac{2 x^{3}}{3 x^{7} + 2})$

Step 1.4.5.4

Multiply $\frac{2 x^{3}}{3 x^{7} + 2}$ by $1$ .

$f (- x) = \frac{2 x^{3}}{3 x^{7} + 2}$

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 $\frac{2 x^{3}}{3 x^{7} + 2}$ $\neq$ $\frac{2 x^{3}}{3 x^{7} - 2}$ , 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 $\frac{2 x^{3}}{3 x^{7} - 2}$ .

$- f (x) = - \frac{2 x^{3}}{3 x^{7} - 2}$

Step 3.2

Since $\frac{2 x^{3}}{3 x^{7} + 2}$ $\neq$ $- \frac{2 x^{3}}{3 x^{7} - 2}$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5