Algebra Examples

$f (x) = 2 \sec (2 (x - π))$

Step 1

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

$2 (x - π) = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $2 (x - π) = \frac{π}{2} + π n$ by $2$ and simplify.

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Step 2.1.1

Divide each term in $2 (x - π) = \frac{π}{2} + π n$ by $2$ .

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

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

Step 2.1.2.1.2

Divide $x - π$ by $1$ .

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

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 2.1.3.1.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

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

Step 2.1.3.1.2.2

Multiply $2$ by $2$ .

$x - π = \frac{π}{4} + \frac{π n}{2}$

Step 2.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.2.1

Add $π$ to both sides of the equation.

$x = \frac{π}{4} + \frac{π n}{2} + π$

Step 2.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.2.3

Combine $π$ and $\frac{4}{4}$ .

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Add $π$ and $π \cdot 4$ .

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Step 2.2.5.1

Reorder $π$ and $4$ .

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

Step 2.2.5.2

Add $π$ and $4 \cdot π$ .

$x = \frac{π n}{2} + \frac{5 π}{4}$

Step 3

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

Set-Builder Notation:

${x | x \neq \frac{π n}{2} + \frac{5 π}{4}}$ , 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:

$(- \infty, - 2] \cup [2, \infty)$

Set-Builder Notation:

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

Step 5

Determine the domain and range.

Domain: ${x | x \neq \frac{π n}{2} + \frac{5 π}{4}}$ , for any integer $n$

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

Step 6