Find the Domain and Range y=tan(pi/6x)

$y = \tan (\frac{π}{6} x)$

Set the argument in $\tan (\frac{π}{6} x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\frac{π}{6} x = \frac{π}{2} + π n$ , for any integer $n$

Solve for $x$ .

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} (\frac{π}{6} x) = \frac{6}{π} (\frac{π}{2} + π n)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{6}{π} (\frac{π}{6} x)$ .

Combine $\frac{π}{6}$ and $x$ .

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} (\frac{π}{2} + π n)$

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} (\frac{π}{2} + π n)$

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{6}{π} (\frac{π}{2} + π n)$

Cancel the common factor of $π$ .

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{6}{π} (\frac{π}{2} + π n)$

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{6}{π} (\frac{π}{2} + π n)$

Rewrite the expression.

$x = \frac{6}{π} (\frac{π}{2} + π n)$

Simplify the right side.

Simplify $\frac{6}{π} (\frac{π}{2} + π n)$ .

Apply the distributive property.

$x = \frac{6}{π} \cdot \frac{π}{2} + \frac{6}{π} (π n)$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$x = \frac{2 (3)}{π} \cdot \frac{π}{2} + \frac{6}{π} (π n)$

Cancel the common factor.

$x = \frac{2 \cdot 3}{π} \cdot \frac{π}{2} + \frac{6}{π} (π n)$

Rewrite the expression.

$x = \frac{3}{π} π + \frac{6}{π} (π n)$

Cancel the common factor of $π$ .

Cancel the common factor.

$x = \frac{3}{π} π + \frac{6}{π} (π n)$

Rewrite the expression.

$x = 3 + \frac{6}{π} (π n)$

Cancel the common factor of $π$ .

Factor $π$ out of $π n$ .

$x = 3 + \frac{6}{π} (π (n))$

Cancel the common factor.

$x = 3 + \frac{6}{π} (π n)$

Rewrite the expression.

$x = 3 + 6 n$

Reorder $3$ and $6 n$ .

$x = 6 n + 3$

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

Set-Builder Notation:

${x | x \neq 6 n + 3}$ , for any integer $n$

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Determine the domain and range.

Domain: ${x | x \neq 6 n + 3}$ , for any integer $n$

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

Find the Domain and Range y=tan(pi/6x)

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