Question

determine the domain of the function $f(x) = -dfrac{8x}{x^2 - 5x - 36}$. enter your answer in interval notation. domain

Explanation:

Step1: Find when denominator is zero

To find the domain, we need to determine the values of \( x \) for which the denominator \( x^2 - 5x - 36 \) is not zero. First, solve \( x^2 - 5x - 36 = 0 \). Factor the quadratic: \( x^2 - 5x - 36=(x - 9)(x + 4)=0 \).

Step2: Solve for x

Setting each factor equal to zero gives \( x - 9 = 0 \) or \( x + 4 = 0 \), so \( x = 9 \) or \( x=-4 \). These are the values to exclude from the domain.

Step3: Write domain in interval notation

The domain is all real numbers except \( x = - 4 \) and \( x = 9 \). In interval notation, this is \( (-\infty, - 4)\cup(-4,9)\cup(9,\infty) \).

Answer:

\((-\infty, -4) \cup (-4, 9) \cup (9, \infty)\)