Find the domain of the function. \( g(x)=\frac{x}{x^{2}+12 x+32} \) What is the domain of the function \( g(x) \) ? A. \( \{x \mid x \) is a real number \( \} \) c. \( \{x \mid x \neq-4 \) and \( x \neq-8) \)

To find the domain of the function \( g(x) = \frac{x}{x^{2} + 12x + 32} \), we need to determine where the function is defined. The function is undefined whenever the denominator equals zero. So, we first set the denominator equal to zero: \[ x^{2} + 12x + 32 = 0 \] Factoring the quadratic gives us \( (x + 4)(x + 8) = 0 \). Thus, \( x = -4 \) and \( x = -8 \) are the values that make the denominator zero. Therefore, the domain excludes these points, leading us to the conclusion that the domain is \( \{x \mid x \neq -4 \text{ and } x \neq -8\} \). Now, for a little historical context: Rational functions, which include fractions like \( g(x) \), have a history rooted in the development of algebra and calculus over centuries. Mathematicians like Descartes advanced our understanding of functions, paving the way for the formal treatment of domains and ranges we use today! On the practical side, understanding the domain of a function is crucial when applying it in real-life scenarios. For instance, if you were to use \( g(x) \) to model a physical situation—like the flow of a liquid—you'd want to know the constraints on \( x \) to avoid undefined behavior. Being aware of what values to exclude helps avoid errors in calculations and interpretations!