Calculus Examples

f (x) = sec (\frac{π x}{4})

$f (x) = \sec (\frac{π x}{4})$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the argument in

sec (\frac{π x}{4})

$\sec (\frac{π x}{4})$ equal to

\frac{π}{2} + π n

$\frac{π}{2} + π n$ to find where the expression is undefined.

\frac{π x}{4} = \frac{π}{2} + π n

$\frac{π x}{4} = \frac{π}{2} + π n$ , for any integer

n

$n$

Step 1.2

Solve for

x

$x$ .

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Step 1.2.1

Multiply both sides of the equation by

\frac{4}{π}

$\frac{4}{π}$ .

\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (\frac{π}{2} + π n)

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (\frac{π}{2} + π n)$

Step 1.2.2

Simplify both sides of the equation.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Simplify

\frac{4}{π} \cdot \frac{π x}{4}

$\frac{4}{π} \cdot \frac{π x}{4}$ .

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Step 1.2.2.1.1.1

Cancel the common factor of

4

$4$ .

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Step 1.2.2.1.1.1.1

Cancel the common factor.

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (\frac{π}{2} + π n)$

Step 1.2.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{4}{π} (\frac{π}{2} + π n)$

Step 1.2.2.1.1.2

Cancel the common factor of

$π$ .

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Step 1.2.2.1.1.2.1

Factor

$π$ out of

$π x$ .

$\frac{1}{π} (π (x)) = \frac{4}{π} (\frac{π}{2} + π n)$

Step 1.2.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{4}{π} (\frac{π}{2} + π n)$

Step 1.2.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{π} (\frac{π}{2} + π n)$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Simplify

$\frac{4}{π} (\frac{π}{2} + π n)$ .

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Step 1.2.2.2.1.1

Apply the distributive property.

$x = \frac{4}{π} \cdot \frac{π}{2} + \frac{4}{π} (π n)$

Step 1.2.2.2.1.2

Cancel the common factor of

$2$ .

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Step 1.2.2.2.1.2.1

Factor

$2$ out of

$4$ .

$x = \frac{2 (2)}{π} \cdot \frac{π}{2} + \frac{4}{π} (π n)$

Step 1.2.2.2.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 2}{π} \cdot \frac{π}{2} + \frac{4}{π} (π n)$

Step 1.2.2.2.1.2.3

Rewrite the expression.

$x = \frac{2}{π} π + \frac{4}{π} (π n)$

Step 1.2.2.2.1.3

Cancel the common factor of

$π$ .

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Step 1.2.2.2.1.3.1

Cancel the common factor.

$x = \frac{2}{π} π + \frac{4}{π} (π n)$

Step 1.2.2.2.1.3.2

Rewrite the expression.

$x = 2 + \frac{4}{π} (π n)$

Step 1.2.2.2.1.4

Cancel the common factor of

$π$ .

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Step 1.2.2.2.1.4.1

Factor

$π$ out of

$π n$ .

$x = 2 + \frac{4}{π} (π (n))$

Step 1.2.2.2.1.4.2

Cancel the common factor.

$x = 2 + \frac{4}{π} (π n)$

Step 1.2.2.2.1.4.3

Rewrite the expression.

$x = 2 + 4 n$

Step 1.2.3

Reorder

$2$ and

$4 n$ .

$x = 4 n + 2$

Step 1.3

The domain is all values of

$x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq 4 n + 2}$ , for any integer

$n$

Set-Builder Notation:

${x | x \neq 4 n + 2}$ , for any integer

$n$

Step 2

Since the domain is not all real numbers,

$\sec (\frac{π x}{4})$ is not continuous over all real numbers.

Not continuous

Step 3