Precalculus Examples

$\frac{x^{2} - x - 6}{x - 4}$

Step 1

Set the denominator in $\frac{x^{2} - x - 6}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 2

Add $4$ to both sides of the equation.

$x = 4$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 4}$

Step 4

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

Interval Notation:

$(- \infty, 7 - 2 \sqrt{6}] \cup [7 + 2 \sqrt{6}, \infty)$

Set-Builder Notation:

${y | y \leq 7 - 2 \sqrt{6}, y \geq 7 + 2 \sqrt{6}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 4) \cup (4, \infty), {x | x \neq 4}$

Range: $(- \infty, 7 - 2 \sqrt{6}] \cup [7 + 2 \sqrt{6}, \infty), {y | y \leq 7 - 2 \sqrt{6}, y \geq 7 + 2 \sqrt{6}}$

Step 6