Problem 1 - College Algebra

Question

Find the domain of each rational function. $$f(x)=\frac{5 x}{x-4}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\frac{5 x}{x-4}\) is \((-\infty, 4) \cup (4, +\infty)\).

Step 1: Identify the Denominator

The first step is to identify the denominator in the rational function. In the function \(f(x)=\frac{5 x}{x-4}\), the denominator is \(x-4\).

Step 2: Solve for x when the Denominator Equals Zero

Setting the denominator equal to zero and solving for \(x\), we equate \(x-4 = 0\). When solved, we get \(x = 4\). So, \(x = 4\) is the value that makes the denominator of our rational function zero.

Step 3: Define the Domain of the Function

The domain of the function is all valid values of \(x\) for which the function is defined. Since our function is undefined for \(x = 4\), the domain of the function is all other real numbers. Hence, our domain in interval notation is \((-\infty, 4) \cup (4, +\infty)\).