Find the Domain and Range y=3/2csc(pix)

$y = \frac{3}{2} \csc (π x)$

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

$π x = π n$ , for any integer $n$

Divide each term in $π x = π n$ by $π$ and simplify.

Divide each term in $π x = π n$ by $π$ .

$\frac{π x}{π} = \frac{π n}{π}$

Simplify the left side.

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π x}{π} = \frac{π n}{π}$

Divide $x$ by $1$ .

$x = \frac{π n}{π}$

Simplify the right side.

Cancel the common factor of $π$ .

Cancel the common factor.

$x = \frac{π n}{π}$

Divide $n$ by $1$ .

$x = n$

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

Set-Builder Notation:

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

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - \frac{3}{2}] \cup [\frac{3}{2}, \infty)$

Set-Builder Notation:

${y | y \leq - \frac{3}{2}, y \geq \frac{3}{2}}$

Determine the domain and range.

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

Range: $(- \infty, - \frac{3}{2}] \cup [\frac{3}{2}, \infty), {y | y \leq - \frac{3}{2}, y \geq \frac{3}{2}}$