Find the Domain and Range (2x^2-8x-24)/(x^2-36)

$\frac{2 x^{2} - 8 x - 24}{x^{2} - 36}$

Step 1

Set the denominator in $\frac{2 x^{2} - 8 x - 24}{x^{2} - 36}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 36 = 0$

Step 2

Solve for $x$ .

Add $36$ to both sides of the equation.

$x^{2} = 36$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{36}$

Simplify $\pm \sqrt{36}$ .

Rewrite $36$ as $6^{2}$ .

$x = \pm \sqrt{6^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 6$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 6$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 6$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 6, - 6$

Step 3

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

Interval Notation:

$(- \infty, - 6) \cup (- 6, 6) \cup (6, \infty)$

Set-Builder Notation:

${x | x \neq 6, - 6}$

Step 4

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

Interval Notation:

$(- \infty, \frac{4}{3}) \cup (\frac{4}{3}, 2) \cup (2, \infty)$

Set-Builder Notation:

${y | y \neq \frac{4}{3}, 2}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 6) \cup (- 6, 6) \cup (6, \infty), {x | x \neq 6, - 6}$

Range: $(- \infty, \frac{4}{3}) \cup (\frac{4}{3}, 2) \cup (2, \infty), {y | y \neq \frac{4}{3}, 2}$

Step 6