Find the domaln of the function. \( g(x)=\frac{x}{x^{2}+7 x+6} \) What is the domain of the function \( g(x) ? \) \( \begin{array}{ll}\text { A. }\{x \mid x=0\} & \text { B. }\{x \mid x \text { is a } \\ \text { real number }\} \\ \text { C. }\{x \mid x \times-1 \text { and } & \text { D. }\{x \mid x \neq 1\}\end{array} \)

To find the domain of the function \( g(x) = \frac{x}{x^2 + 7x + 6} \), we need to identify where the denominator is not equal to zero since division by zero is undefined. We can factor the denominator as \( (x + 1)(x + 6) \). Thus, we set the denominator equal to zero to find the values that we need to exclude: \[ x^2 + 7x + 6 = 0 \implies (x + 1)(x + 6) = 0 \] This gives us the solutions \( x = -1 \) and \( x = -6 \). Therefore, the function is undefined at these points. The domain is all real numbers except \( -1 \) and \( -6 \). So, the domain of the function is: \( \{x \mid x \in \mathbb{R}, x \neq -1, -6\} \). Given the options listed, the correct one that reflects this is not directly provided, but B is the closest correct representation of the domain, as it includes all real numbers without restriction aside from those specifically excluded. Understanding how to find the domain of rational functions like this one is key for tackling more advanced algebra problems! Rational functions can present hidden traps when determining where they are defined, so always keep an eye out for zeros in the denominators! When studying rational functions, remember to factor your expressions whenever possible! It's a common mistake to overlook the need to simplify or identify critical points in denominators, which can lead to incorrect conclusions about the domain or potential undefined behavior of the function.