Precalculus Examples

$f (x) = \frac{x^{2} + 9 x - 9}{x - 9}$

Step 1

Set the denominator in $\frac{x^{2} + 9 x - 9}{x - 9}$ equal to $0$ to find where the expression is undefined.

$x - 9 = 0$

Step 2

Add $9$ to both sides of the equation.

$x = 9$

Step 3

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

Interval Notation:

$(- \infty, 9) \cup (9, \infty)$

Set-Builder Notation:

${x | x \neq 9}$

Step 4

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

Interval Notation:

$(- \infty, 27 - 6 \sqrt{17}] \cup [27 + 6 \sqrt{17}, \infty)$

Set-Builder Notation:

${y | y \leq 27 - 6 \sqrt{17}, y \geq 27 + 6 \sqrt{17}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 9) \cup (9, \infty), {x | x \neq 9}$

Range: $(- \infty, 27 - 6 \sqrt{17}] \cup [27 + 6 \sqrt{17}, \infty), {y | y \leq 27 - 6 \sqrt{17}, y \geq 27 + 6 \sqrt{17}}$

Step 6