Problem 54 - College Algebra

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 \( x \in \mathbb{R}, x \neq -5 \), or in interval notation, \( (-\infty, -5) \cup (-5, +\infty) \).

Step 1: Identify the function

The given function is \( g(x) = \frac{2}{x+5} \). We want to find the values for x that are permissible.

Step 2: Determine when the denominator equals zero

For the function \( g(x) = \frac{2}{x+5} \), the denominator of the fraction cannot be equal to zero since division by zero is undefined. Therefore, we shall find the value of x for which the denominator \( x + 5 = 0 \). So, \( x = -5 \).

Step 3: Determine the domain of the function

Every real number is permitted, except for \(x = -5\), which would make the denominator of the function equal to zero. Hence, the domain of the function \( g(x) = \frac{2}{x+5} \) is \( x \in \mathbb{R}, x \neq -5 \), or in interval notation, \( (-\infty, -5) \cup (-5, +\infty) \).