Find the Domain and Range y=tan(pi/9x)

$y = \tan (\frac{π}{9} x)$

Set the argument in $\tan (\frac{π}{9} x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\frac{π}{9} x = \frac{π}{2} + π n$ , for any integer $n$

Solve for $x$ .

Multiply both sides of the equation by $\frac{9}{π}$ .

$\frac{9}{π} (\frac{π}{9} x) = \frac{9}{π} (\frac{π}{2} + π n)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{9}{π} (\frac{π}{9} x)$ .

Combine $\frac{π}{9}$ and $x$ .

$\frac{9}{π} \cdot \frac{π x}{9} = \frac{9}{π} (\frac{π}{2} + π n)$

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9}{π} \cdot \frac{π x}{9} = \frac{9}{π} (\frac{π}{2} + π n)$

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{9}{π} (\frac{π}{2} + π n)$

Cancel the common factor of $π$ .

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{9}{π} (\frac{π}{2} + π n)$

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{9}{π} (\frac{π}{2} + π n)$

Rewrite the expression.

$x = \frac{9}{π} (\frac{π}{2} + π n)$

Simplify the right side.

Simplify $\frac{9}{π} (\frac{π}{2} + π n)$ .

Apply the distributive property.

$x = \frac{9}{π} \cdot \frac{π}{2} + \frac{9}{π} (π n)$

Cancel the common factor of $π$ .

Cancel the common factor.

$x = \frac{9}{π} \cdot \frac{π}{2} + \frac{9}{π} (π n)$

Rewrite the expression.

$x = 9 (\frac{1}{2}) + \frac{9}{π} (π n)$

Combine $9$ and $\frac{1}{2}$ .

$x = \frac{9}{2} + \frac{9}{π} (π n)$

Cancel the common factor of $π$ .

Factor $π$ out of $π n$ .

$x = \frac{9}{2} + \frac{9}{π} (π (n))$

Cancel the common factor.

$x = \frac{9}{2} + \frac{9}{π} (π n)$

Rewrite the expression.

$x = \frac{9}{2} + 9 n$

Reorder $\frac{9}{2}$ and $9 n$ .

$x = 9 n + \frac{9}{2}$

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

Set-Builder Notation:

${x | x \neq 9 n + \frac{9}{2}}$ , 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: ${x | x \neq 9 n + \frac{9}{2}}$ , for any integer $n$

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

Find the Domain and Range y=tan(pi/9x)

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