Algebra Examples

$y = - \frac{8}{27} \sec (\frac{π x}{12} - \sqrt{9}) - 28$

Step 1

Set the argument in $\sec (\frac{π x}{12} - \sqrt{9})$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\frac{π x}{12} - \sqrt{9} = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $9$ as $3^{2}$ .

$\frac{π x}{12} - \sqrt{3^{2}} = \frac{π}{2} + π n$

Step 2.1.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{π x}{12} - 1 \cdot 3 = \frac{π}{2} + π n$

Step 2.1.3

Multiply $- 1$ by $3$ .

$\frac{π x}{12} - 3 = \frac{π}{2} + π n$

Step 2.2

Add $3$ to both sides of the equation.

$\frac{π x}{12} = \frac{π}{2} + π n + 3$

Step 2.3

Multiply both sides of the equation by $\frac{12}{π}$ .

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} (\frac{π}{2} + π n + 3)$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify $\frac{12}{π} \cdot \frac{π x}{12}$ .

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Step 2.4.1.1.1

Cancel the common factor of $12$ .

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Step 2.4.1.1.1.1

Cancel the common factor.

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} (\frac{π}{2} + π n + 3)$

Step 2.4.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{12}{π} (\frac{π}{2} + π n + 3)$

Step 2.4.1.1.2

Cancel the common factor of $π$ .

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Step 2.4.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{12}{π} (\frac{π}{2} + π n + 3)$

Step 2.4.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{12}{π} (\frac{π}{2} + π n + 3)$

Step 2.4.1.1.2.3

Rewrite the expression.

$x = \frac{12}{π} (\frac{π}{2} + π n + 3)$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\frac{12}{π} (\frac{π}{2} + π n + 3)$ .

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Step 2.4.2.1.1

Apply the distributive property.

$x = \frac{12}{π} \cdot \frac{π}{2} + \frac{12}{π} (π n) + \frac{12}{π} \cdot 3$

Step 2.4.2.1.2

Simplify.

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Step 2.4.2.1.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.1.2.1.1

Factor $2$ out of $12$ .

$x = \frac{2 (6)}{π} \cdot \frac{π}{2} + \frac{12}{π} (π n) + \frac{12}{π} \cdot 3$

Step 2.4.2.1.2.1.2

Cancel the common factor.

$x = \frac{2 \cdot 6}{π} \cdot \frac{π}{2} + \frac{12}{π} (π n) + \frac{12}{π} \cdot 3$

Step 2.4.2.1.2.1.3

Rewrite the expression.

$x = \frac{6}{π} π + \frac{12}{π} (π n) + \frac{12}{π} \cdot 3$

Step 2.4.2.1.2.2

Cancel the common factor of $π$ .

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Step 2.4.2.1.2.2.1

Cancel the common factor.

$x = \frac{6}{π} π + \frac{12}{π} (π n) + \frac{12}{π} \cdot 3$

Step 2.4.2.1.2.2.2

Rewrite the expression.

$x = 6 + \frac{12}{π} (π n) + \frac{12}{π} \cdot 3$

Step 2.4.2.1.2.3

Cancel the common factor of $π$ .

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Step 2.4.2.1.2.3.1

Factor $π$ out of $π n$ .

$x = 6 + \frac{12}{π} (π (n)) + \frac{12}{π} \cdot 3$

Step 2.4.2.1.2.3.2

Cancel the common factor.

$x = 6 + \frac{12}{π} (π n) + \frac{12}{π} \cdot 3$

Step 2.4.2.1.2.3.3

Rewrite the expression.

$x = 6 + 12 n + \frac{12}{π} \cdot 3$

Step 2.4.2.1.2.4

Multiply $\frac{12}{π} \cdot 3$ .

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Step 2.4.2.1.2.4.1

Combine $\frac{12}{π}$ and $3$ .

$x = 6 + 12 n + \frac{12 \cdot 3}{π}$

Step 2.4.2.1.2.4.2

Multiply $12$ by $3$ .

$x = 6 + 12 n + \frac{36}{π}$

Step 2.5

Reorder $6$ and $12 n$ .

$x = 12 n + 6 + \frac{36}{π}$

Step 3

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

Set-Builder Notation:

${x | x \neq 12 n + 6 + \frac{36}{π}}$ , 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:

$(- \infty, - \frac{764}{27}] \cup [- \frac{748}{27}, \infty)$

Set-Builder Notation:

${y | y \leq - \frac{764}{27}, y \geq - \frac{748}{27}}$

Step 5

Determine the domain and range.

Domain: ${x | x \neq 12 n + 6 + \frac{36}{π}}$ , for any integer $n$

Range: $(- \infty, - \frac{764}{27}] \cup [- \frac{748}{27}, \infty), {y | y \leq - \frac{764}{27}, y \geq - \frac{748}{27}}$

Step 6