Algebra Examples

$y = \sec^{-1} (x)$

Step 1

Set the argument in $\sec^{-1} (x)$ less than or equal to $- 1$ to find where the expression is defined.

$x \leq - 1$

Step 2

Set the argument in $\sec^{-1} (x)$ greater than or equal to $1$ to find where the expression is defined.

$x \geq 1$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \leq - 1, x \geq 1}$

Step 4

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

Interval Notation:

$[0, \frac{π}{2}) \cup (\frac{π}{2}, π]$

Set-Builder Notation:

${y | 0 \leq y \leq π, y \neq \frac{π}{2}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 1] \cup [1, \infty), {x | x \leq - 1, x \geq 1}$

Range: $[0, \frac{π}{2}) \cup (\frac{π}{2}, π], {y | 0 \leq y \leq π, y \neq \frac{π}{2}}$

Step 6