Find the Domain and Range g(x)=-2sec(3x+4)+1

$g (x) = - 2 \sec (3 x + 4) + 1$

Step 1

Set the argument in $\sec (3 x + 4)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$3 x + 4 = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

Subtract $4$ from both sides of the equation.

$3 x = \frac{π}{2} + π n - 4$

Divide each term in $3 x = \frac{π}{2} + π n - 4$ by $3$ and simplify.

Divide each term in $3 x = \frac{π}{2} + π n - 4$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{3} + \frac{π n}{3} + \frac{- 4}{3}$

Multiply $\frac{π}{2} \cdot \frac{1}{3}$ .

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

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

Multiply $2$ by $3$ .

$x = \frac{π}{6} + \frac{π n}{3} + \frac{- 4}{3}$

Move the negative in front of the fraction.

$x = \frac{π}{6} + \frac{π n}{3} - \frac{4}{3}$

Step 3

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

Set-Builder Notation:

${x | x \neq \frac{π}{6} + \frac{π n}{3} - \frac{4}{3}}$ , 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:

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

Set-Builder Notation:

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

Step 5

Determine the domain and range.

Domain: ${x | x \neq \frac{π}{6} + \frac{π n}{3} - \frac{4}{3}}$ , for any integer $n$

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

Step 6