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 values of \( x \) that would make the denominator equal to zero. The expression \( x^{2} + 7x + 6 \) can be factored as \( (x + 1)(x + 6) \). Therefore, the denominator is equal to zero when \( x = -1 \) or \( x = -6 \). Consequently, the domain of \( g(x) \) includes all real numbers except \( -1 \) and \( -6 \): In set notation, this can be described as \( \{ x \mid x \in \mathbb{R}, x \neq -1, x \neq -6 \} \). So, since none of the options provided seem to match this conclusion directly, the correct answer would encompass all real numbers excluding \( -1 \) and \( -6 \). Here's some extra tidbit: Did you know that rational functions, like the one in this question, often have interesting behaviors around their vertical asymptotes? The points we exclude from the domain (where the denominator equals zero) create vertical asymptotes on the graph of the function, which can lead to fascinating shapes and offsets! Moreover, while tackling functions and their domains, it's good to remember that such exclusions help us to avoid undefined scenarios in mathematics. By spotting these values early on, you can save a lot of hassle and seamlessly navigate through more complex functions later!