Find the Domain and Range y=3sec((2x)/3)

$y = 3 \sec (\frac{2 x}{3})$

Step 1

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

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

Step 2

Solve for $x$ .

Multiply both sides of the equation by $\frac{3}{2}$ .

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{3}{2} \cdot \frac{2 x}{3}$ .

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{2} (2 x) = \frac{3}{2} (\frac{π}{2} + π n)$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

$\frac{1}{2} (2 (x)) = \frac{3}{2} (\frac{π}{2} + π n)$

Cancel the common factor.

$\frac{1}{2} (2 x) = \frac{3}{2} (\frac{π}{2} + π n)$

Rewrite the expression.

$x = \frac{3}{2} (\frac{π}{2} + π n)$

Simplify the right side.

Simplify $\frac{3}{2} (\frac{π}{2} + π n)$ .

Apply the distributive property.

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

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

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

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

Multiply $2$ by $2$ .

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

Multiply $\frac{3}{2} (π n)$ .

Combine $π$ and $\frac{3}{2}$ .

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

Combine $\frac{π \cdot 3}{2}$ and $n$ .

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

Move $3$ to the left of $π$ .

$x = \frac{3 π}{4} + \frac{3 π n}{2}$

Reorder $\frac{3 π}{4}$ and $\frac{3 π n}{2}$ .

$x = \frac{3 π n}{2} + \frac{3 π}{4}$

Step 3

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

Set-Builder Notation:

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

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

Set-Builder Notation:

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

Step 5

Determine the domain and range.

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

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

Step 6

Find the Domain and Range y=3sec((2x)/3)

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