Algebra Examples

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

Step 1

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 $n$

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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Step 2.1.2.1

Combine the numerators over the common denominator.

$2 x = π n + \frac{π - π}{2}$

Step 2.1.2.2

Subtract $π$ from $π$ .

$2 x = π n + \frac{0}{2}$

Step 2.1.3

Divide $0$ by $2$ .

$2 x = π n + 0$

Step 2.1.4

Add $π n$ and $0$ .

$2 x = π n$

Step 2.2

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

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Step 2.2.1

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

$\frac{2 x}{2} = \frac{π n}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π n}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π n}{2}$

Step 3

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 $n$

Step 4

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:

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

Step 5

Determine the domain and range.

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

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

Step 6