Precalculus Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 2

Find $f (- x)$ .

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Step 2.1

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

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

Step 2.2

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Simplify the expression.

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Step 2.10.1

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

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

Step 2.10.2

Multiply $- 1$ by $- 1$ .

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

Step 2.10.3

Multiply $x - 1$ by $1$ .

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

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 $- \frac{x}{(x - 1) (x + 1)}$ $\neq$ $\frac{x}{(x + 1) (x - 1)}$ , 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 $- 1$ by $\frac{x}{(x + 1) (x - 1)}$ .

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

Step 4.2

Since $- \frac{x}{(x - 1) (x + 1)} = - \frac{x}{(x + 1) (x - 1)}$ , the function is odd.

The function is odd

Step 5