Algebra Examples

$\frac{\sqrt{x - 2}}{x - 3}$

Step 1

Set the radicand in $\sqrt{x - 2}$ greater than or equal to $0$ to find where the expression is defined.

$x - 2 \geq 0$

Step 2

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 3

Set the denominator in $\frac{\sqrt{x - 2}}{x - 3}$ equal to $0$ to find where the expression is undefined.

$x - 3 = 0$

Step 4

Add $3$ to both sides of the equation.

$x = 3$

Step 5

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

Interval Notation:

$[2, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x \geq 2, x \neq 3}$

Step 6

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 7

Determine the domain and range.

Domain: $[2, 3) \cup (3, \infty), {x | x \geq 2, x \neq 3}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 8