Algebra Examples

$y = \sec (x + π)$

Step 1

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

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

Step 2

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

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Step 2.1

Subtract $π$ from both sides of the equation.

$x = \frac{π}{2} + π n - π$

Step 2.2

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3

Combine $- π$ and $\frac{2}{2}$ .

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

Step 2.4

Combine the numerators over the common denominator.

$x = π n + \frac{π - π \cdot 2}{2}$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Multiply $2$ by $- 1$ .

$x = π n + \frac{π - 2 π}{2}$

Step 2.5.1.2

Subtract $2 π$ from $π$ .

$x = π n + \frac{- π}{2}$

Step 2.5.2

Move the negative in front of the fraction.

$x = π n - \frac{π}{2}$

Step 3

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

Set-Builder Notation:

${x | x \neq π n - \frac{π}{2}}$ , 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:

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

Set-Builder Notation:

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

Step 5

Determine the domain and range.

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

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

Step 6