Find the Domain and Range y=tan(2x-pi)

$y = \tan (2 x - π)$

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

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

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Add $π$ to both sides of the equation.

$2 x = \frac{π}{2} + π n + π$

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$2 x = π n + \frac{π + π \cdot 2}{2}$

Add $π$ and $π \cdot 2$ .

Reorder $π$ and $2$ .

$2 x = π n + \frac{π + 2 \cdot π}{2}$

Add $π$ and $2 \cdot π$ .

$2 x = π n + \frac{3 π}{2}$

Divide each term in $2 x = π n + \frac{3 π}{2}$ by $2$ and simplify.

Divide each term in $2 x = π n + \frac{3 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{π n}{2} + \frac{\frac{3 π}{2}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{π n}{2} + \frac{\frac{3 π}{2}}{2}$

Divide $x$ by $1$ .

$x = \frac{π n}{2} + \frac{\frac{3 π}{2}}{2}$

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π n}{2} + \frac{3 π}{2} \cdot \frac{1}{2}$

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

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

Multiply $2$ by $2$ .

$x = \frac{π n}{2} + \frac{3 π}{4}$

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

Set-Builder Notation:

${x | x \neq \frac{π n}{2} + \frac{3 π}{4}}$ , 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: ${x | x \neq \frac{π n}{2} + \frac{3 π}{4}}$ , for any integer $n$

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

Find the Domain and Range y=tan(2x-pi)

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