Precalculus Examples

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

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

Step 1

Simplify the denominator.

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Step 1.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 1

$b = 1$ .

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

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

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

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

Step 2

Find

f (- x)

$f (- x)$ .

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Step 2.1

Find

f (- x)

$f (- x)$ by substituting

- x

$- x$ for all occurrence of

x

$x$ in

f (x)

$f (x)$ .

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

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

Step 2.2

Move the negative in front of the fraction.

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

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

Step 2.3

Factor

- 1

$- 1$ out of

- x

$- x$ .

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

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

Step 2.4

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

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

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