Find the Domain and Range y=csc(1/2x)

$y = \csc (\frac{1}{2} x)$

Step 1

Set the argument in $\csc (\frac{1}{2} x)$ equal to $π n$ to find where the expression is undefined.

$\frac{1}{2} x = π n$ , for any integer $n$

Step 2

Solve for $x$ .

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} x) = 2 (π n)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $2 (\frac{1}{2} x)$ .

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

$2 \frac{x}{2} = 2 (π n)$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{x}{2} = 2 (π n)$

Rewrite the expression.

$x = 2 (π n)$

Simplify the right side.

Remove parentheses.

$x = 2 π n$

Step 3

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

Set-Builder Notation:

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

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

Step 6