Algebra Examples

$y = \sec (x - \frac{π}{2})$

Step 1

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

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

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{π}{2} + π n + \frac{π}{2}$

Step 2.2

Combine the numerators over the common denominator.

$x = π n + \frac{π + π}{2}$

Step 2.3

Add $π$ and $π$ .

$x = π n + \frac{2 π}{2}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$x = π n + \frac{2 π}{2}$

Step 2.4.2

Divide $π$ by $1$ .

$x = π n + π$

Step 3

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

Set-Builder Notation:

${x | x \neq π n + π}$ , 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:

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

Set-Builder Notation:

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

Step 5

Determine the domain and range.

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

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

Step 6