Precalculus Examples

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

Step 1

Find $f (- x)$ .

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Step 1.1

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

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

Step 1.2

Move the negative in front of the fraction.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Simplify the expression.

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Step 1.6.1

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

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

Step 1.6.2

Move the negative in front of the fraction.

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

Step 1.6.3

Multiply $- 1$ by $- 1$ .

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

Step 1.6.4

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

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

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

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

Step 3.2

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

The function is not odd

Step 4

The function is neither odd nor even

Step 5