Precalculus Examples

$f (x) = x + \frac{12}{x}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = (- x) + \frac{12}{- x}$

Step 1.2

Move the negative in front of the fraction.

$f (- x) = - x - \frac{12}{x}$

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 $- x - \frac{12}{x}$ $\neq$ $x + \frac{12}{x}$ , 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 $x + \frac{12}{x}$ by $- 1$ .

$- f (x) = - (x + \frac{12}{x})$

Step 3.1.2

Apply the distributive property.

$- f (x) = - x - \frac{12}{x}$

Step 3.2

Since $- x - \frac{12}{x} = - x - \frac{12}{x}$ , the function is odd.

The function is odd

Step 4