Precalculus Examples

$y = \frac{\cos (x)}{x}$

Step 1

Write $y = \frac{\cos (x)}{x}$ as a function.

$f (x) = \frac{\cos (x)}{x}$

Step 2

Find $f (- x)$ .

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Step 2.1

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

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

Step 2.2

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

Step 2.3

Move the negative in front of the fraction.

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

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 $- \frac{\cos (x)}{x}$ $\neq$ $\frac{\cos (x)}{x}$ , 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 $\frac{\cos (x)}{x}$ .

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

Step 4.2

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

The function is odd

Step 5