Question

question 2
find the domain of the function using interval notation.
$f(x) = \frac{1}{x^2 - 2x - 8}$

Explanation:

Step1: Identify undefined condition

A rational function is undefined when its denominator equals 0. Set the denominator to 0:
$x^2 - 2x - 8 = 0$

Step2: Factor the quadratic equation

Factor the polynomial to solve for $x$:
$(x - 4)(x + 2) = 0$

Step3: Solve for x values

Set each factor equal to 0 and solve:
$x - 4 = 0 \implies x = 4$
$x + 2 = 0 \implies x = -2$

Step4: Define domain in intervals

The function is defined for all real numbers except $x=-2$ and $x=4$. Express this in interval notation:
$(-\infty, -2) \cup (-2, 4) \cup (4, \infty)$

Answer:

$(-\infty, -2) \cup (-2, 4) \cup (4, \infty)$