Problem 4 - College Algebra Essentials

Question

Find the domain of each function. $$g(x)=\frac{2}{x+5}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \frac{2}{x+5}\) is all real numbers except \(x = -5\).

Step 1: Identify the Rule

For a function \(y=f(x)\), division by zero is not allowed. That is, for any function \(y=\frac{a}{g(x)}\), \(g(x)\neq 0\). In our function \(g(x) =\frac{2}{x+5}\), to find its domain, we look for the values that would make the denominator zero, i.e., \(x+5 = 0\).

Step 2: Solve the Equation

To find the values, solve the equation \(x+5=0\). Here, \(x = -5\).

Step 3: Express the Domain

The domain of the function is all real numbers except \(x = -5\) as it would make the denominator equal to zero. Thus, the domain is \(x \in (-\infty, -5) \cup (-5, +\infty)\), which represents all real numbers except -5.