Find the Domain and Range y=4sec(1/2x)

$y = 4 \sec (\frac{1}{2} x)$

Step 1

Set the argument in $\sec (\frac{1}{2} x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\frac{1}{2} x = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

Multiply both sides of the equation by $2$ .

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $2 (\frac{1}{2} x)$ .

Combine $\frac{1}{2}$ and $x$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = 2 (\frac{π}{2} + π n)$

Simplify the right side.

Simplify $2 (\frac{π}{2} + π n)$ .

Apply the distributive property.

$x = 2 \frac{π}{2} + 2 (π n)$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = 2 \frac{π}{2} + 2 (π n)$

Rewrite the expression.

$x = π + 2 (π n)$

$x = π + 2 π n$

Reorder $π$ and $2 π n$ .

$x = 2 π n + π$

Step 3

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

Set-Builder Notation:

${x | x \neq 2 π n + π}$ , 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:

$(- \infty, - 4] \cup [4, \infty)$

Set-Builder Notation:

${y | y \leq - 4, y \geq 4}$

Step 5

Determine the domain and range.

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

Range: $(- \infty, - 4] \cup [4, \infty), {y | y \leq - 4, y \geq 4}$

Step 6