Precalculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Separate fractions.

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

Step 2.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 2.4

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 2.5

Divide $x$ by $1$ .

$f (x) = x \cot (x)$

Step 3

Find $f (- x)$ .

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Step 3.1

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

$f (- x) = (- x) \cot (- x)$

Step 3.2

Since $\cot (- x)$ is an odd function, rewrite $\cot (- x)$ as $- \cot (x)$ .

$f (- x) = - x (- \cot (x))$

Step 3.3

Multiply $- x (- \cot (x))$ .

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

$f (- x) = 1 x \cot (x)$

Step 3.3.2

Multiply $x$ by $1$ .

$f (- x) = x \cot (x)$

Step 4

A function is even if $f (- x) = f (x)$ .

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Step 4.1

Check if $f (- x) = f (x)$ .

Step 4.2

Since $x \cot (x) = x \cot (x)$ , the function is even.

The function is even

Step 5