Find the Domain and Range f(x)=sec(x)+1/2

$f (x) = \sec (x) + \frac{1}{2}$

Step 1

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

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

Step 2

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

Set-Builder Notation:

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

Step 3

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - \frac{1}{2}] \cup [\frac{3}{2}, \infty)$

Set-Builder Notation:

${y | y \leq - \frac{1}{2}, y \geq \frac{3}{2}}$

Step 4

Determine the domain and range.

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

Range: $(- \infty, - \frac{1}{2}] \cup [\frac{3}{2}, \infty), {y | y \leq - \frac{1}{2}, y \geq \frac{3}{2}}$

Step 5