Precalculus Examples

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

Step 1

Find $f (- x)$ .

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Step 1.1

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

Cancel the common factor of $1$ and $- 1$ .

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Step 1.2.1.1

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

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

Step 1.2.1.2

Move the negative in front of the fraction.

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

Step 1.2.2

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

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Step 1.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.2.2

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

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

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

Find $- f (x)$ .

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Step 3.1.1

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

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

Step 3.1.2

Apply the distributive property.

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

Step 3.2

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

The function is odd

Step 4