Precalculus Examples

$y = \cot (\frac{1}{12} x)$

Step 1

Set the argument in $\cot (\frac{1}{12} x)$ equal to $π n$ to find where the expression is undefined.

$\frac{1}{12} x = π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $12$ .

$12 (\frac{1}{12} x) = 12 (π n)$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $12 (\frac{1}{12} x)$ .

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Step 2.2.1.1.1

Combine $\frac{1}{12}$ and $x$ .

$12 \frac{x}{12} = 12 (π n)$

Step 2.2.1.1.2

Cancel the common factor of $12$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$12 \frac{x}{12} = 12 (π n)$

Step 2.2.1.1.2.2

Rewrite the expression.

$x = 12 (π n)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Remove parentheses.

$x = 12 π n$

Step 3

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq 12 π n}$ , for any integer $n$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: ${x | x \neq 12 π n}$ , for any integer $n$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6