Find the Domain and Range y=csc(x+pi/4)

$y = \csc (x + \frac{π}{4})$

Step 1

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

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

Step 2

Subtract $\frac{π}{4}$ from both sides of the equation.

$x = π n - \frac{π}{4}$

Step 3

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

Set-Builder Notation:

${x | x \neq π n - \frac{π}{4}}$ , 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:

$(- \infty, - 1] \cup [1, \infty)$

Set-Builder Notation:

${y | y \leq - 1, y \geq 1}$

Step 5

Determine the domain and range.

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

Range: $(- \infty, - 1] \cup [1, \infty), {y | y \leq - 1, y \geq 1}$

Step 6