Find the Domain and Range y=-2sec(2x+pi/2)+7

$y = - 2 \sec (2 x + \frac{π}{2}) + 7$

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

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

Solve for $x$ .

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

Subtract $\frac{π}{2}$ from both sides of the equation.

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

Combine the opposite terms in $\frac{π}{2} + π n - \frac{π}{2}$ .

Combine the numerators over the common denominator.

$2 x = π n + \frac{π - π}{2}$

Subtract $π$ from $π$ .

$2 x = π n + \frac{0}{2}$

Divide $0$ by $2$ .

$2 x = π n + 0$

Add $π n$ and $0$ .

$2 x = π n$

Divide each term in $2 x = π n$ by $2$ and simplify.

Divide each term in $2 x = π n$ by $2$ .

$\frac{2 x}{2} = \frac{π n}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{π n}{2}$

Divide $x$ by $1$ .

$x = \frac{π n}{2}$

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

Set-Builder Notation:

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

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, 5] \cup [9, \infty)$

Set-Builder Notation:

${y | y \leq 5, y \geq 9}$

Determine the domain and range.

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

Range: $(- \infty, 5] \cup [9, \infty), {y | y \leq 5, y \geq 9}$