Find the Domain and Range y=1/4tan(x-pi/4)

$y = \frac{1}{4} \tan (x - \frac{π}{4})$

Set the argument in $\tan (x - \frac{π}{4})$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x - \frac{π}{4} = \frac{π}{2} + π n$ , for any integer $n$

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

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{π}{2} + π n + \frac{π}{4}$

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

$x = π n + \frac{π}{2} \cdot \frac{2}{2} + \frac{π}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

$x = π n + \frac{π \cdot 2}{2 \cdot 2} + \frac{π}{4}$

Multiply $2$ by $2$ .

$x = π n + \frac{π \cdot 2}{4} + \frac{π}{4}$

Combine the numerators over the common denominator.

$x = π n + \frac{π \cdot 2 + π}{4}$

Add $π \cdot 2$ and $π$ .

Reorder $π$ and $2$ .

$x = π n + \frac{2 \cdot π + π}{4}$

Add $2 \cdot π$ and $π$ .

$x = π n + \frac{3 \cdot π}{4}$

$x = π n + \frac{3 π}{4}$

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

Set-Builder Notation:

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

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