Calculus Examples

$arccsc (x)$

Set the argument in $arccsc (x)$ less than or equal to $- 1$ to find where the expression is defined.

$x \leq - 1$

Set the argument in $arccsc (x)$ greater than or equal to $1$ to find where the expression is defined.

$x \geq 1$

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

Interval Notation:

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

Set-Builder Notation:

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

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

Interval Notation:

$[- \frac{π}{2}, 0) \cup (0, \frac{π}{2}]$

Set-Builder Notation:

${y | - \frac{π}{2} \leq y \leq \frac{π}{2}, y \neq 0}$

Determine the domain and range.

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

Range: $[- \frac{π}{2}, 0) \cup (0, \frac{π}{2}], {y | - \frac{π}{2} \leq y \leq \frac{π}{2}, y \neq 0}$