Calculus Examples

$f (x) = 8 \csc (x^{2})$

Step 1

Find the domain to determine if the expression is continuous.

Tap for more steps...

Step 1.1

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

$x^{2} = π n$ , for any integer $n$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{π n}$

Step 1.2.2

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 1.2.2.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{π n}$

Step 1.2.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{π n}$

Step 1.2.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{π n}, - \sqrt{π n}$

Step 1.3

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

Set-Builder Notation:

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

Set-Builder Notation:

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

Step 2

Since the domain is not all real numbers, $8 \csc (x^{2})$ is not continuous over all real numbers.

Not continuous

Step 3