Problem 6 - College Algebra

Question

Find the domain of each rational function. $$h(x)=\frac{x+8}{x^{2}-64}$$

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\frac{x+8}{x^{2}-64}\) is \(x \in (-\infty,-8) \cup (-8,8) \cup (8,\infty)\).

Step 1: Identify the Denominator

In the function \(h(x)=\frac{x+8}{x^{2}-64}\), the denominator is \(x^{2}-64\).

Step 2: Solve the Denominator for Zero

Set the denominator equal to zero and solve for x. This gives the equation \(x^{2}-64 = 0\). Solving this equation yields \(x = 8\) and \(x = -8\).

Step 3: Exclude these Values from the Domain

The domain of a function is all possible x-values, excluding any values that would result in the function being undefined. In this case, since \(x=8\) and \(x=-8\) would cause the denominator of the function to be zero, these values are excluded from the domain. Therefore, the domain is \(x \in (-\infty,-8) \cup (-8,8) \cup (8,\infty) \).