Algebra Examples

$g (x) = - 2 \sec (2 x + 3) - 1$

Step 1

Set the argument in $\sec (2 x + 3)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$2 x + 3 = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $3$ from both sides of the equation.

$2 x = \frac{π}{2} + π n - 3$

Step 2.2

Divide each term in $2 x = \frac{π}{2} + π n - 3$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = \frac{π}{2} + π n - 3$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{π n}{2} + \frac{- 3}{2}$

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.

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{π n}{2} + \frac{- 3}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2} + \frac{π n}{2} + \frac{- 3}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2} + \frac{π n}{2} + \frac{- 3}{2}$

Step 2.2.3.1.2

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

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Step 2.2.3.1.2.1

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

$x = \frac{π}{2 \cdot 2} + \frac{π n}{2} + \frac{- 3}{2}$

Step 2.2.3.1.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4} + \frac{π n}{2} + \frac{- 3}{2}$

Step 2.2.3.1.3

Move the negative in front of the fraction.

$x = \frac{π}{4} + \frac{π n}{2} - \frac{3}{2}$

Step 3

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

Set-Builder Notation:

${x | x \neq \frac{π}{4} + \frac{π n}{2} - \frac{3}{2}}$ , 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:

$(- \infty, - 3] \cup [1, \infty)$

Set-Builder Notation:

${y | y \leq - 3, y \geq 1}$

Step 5

Determine the domain and range.

Domain: ${x | x \neq \frac{π}{4} + \frac{π n}{2} - \frac{3}{2}}$ , for any integer $n$

Range: $(- \infty, - 3] \cup [1, \infty), {y | y \leq - 3, y \geq 1}$

Step 6