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

$j (x) = \tan (\frac{x}{2} - π)$

Step 1

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

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

Step 2

Solve for $x$ .

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

Add $π$ to both sides of the equation.

$\frac{x}{2} = \frac{π}{2} + π n + π$

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Add $π$ and $π \cdot 2$ .

Reorder $π$ and $2$ .

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

Add $π$ and $2 \cdot π$ .

$\frac{x}{2} = π n + \frac{3 π}{2}$

Multiply both sides of the equation by $2$ .

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

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = 2 (π n + \frac{3 π}{2})$

Simplify the right side.

Simplify $2 (π n + \frac{3 π}{2})$ .

Apply the distributive property.

$x = 2 (π n) + 2 \frac{3 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = 2 π n + 2 \frac{3 π}{2}$

Rewrite the expression.

$x = 2 π n + 3 π$

Step 3

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

Set-Builder Notation:

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

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

Step 6

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

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