Find the domain of the function. \( g(x)=\frac{9}{7-6 x} \) Choose the correct domain below. A. \( \{x \mid x \) is a real number and \( x \neq 9\} \) B. \( \{x \mid x \) is a real number and \( x \neq 0\} \) C. \( \left\{x \mid x\right. \) is a real number and \( \left.x \geq \frac{7}{6}\right\} \) D. \( \left\{x \mid x\right. \) is a real number and \( \left.x \neq \frac{7}{6}\right\} \)

To find the domain of the function \( g(x)=\frac{9}{7-6 x} \), we need to determine where the function is defined. The function is undefined when the denominator equals zero. Setting the denominator equal to zero gives us: \( 7 - 6x = 0 \) Solving for \( x \), we get: \( 6x = 7 \) \( x = \frac{7}{6} \) Therefore, the domain of the function is all real numbers except \( x = \frac{7}{6} \). So, the correct answer is **D. \( \left\{x \mid x \text{ is a real number and } x \neq \frac{7}{6}\right\} \)**. The domain of a function is a crucial concept in mathematics, often hinting at the behavior of the function. For instance, undefined points typically suggest vertical asymptotes, leading to interesting graph behavior like going to positive or negative infinity! Understanding domains helps in real-world applications like physics, where certain equations can only be solved within specific ranges. For example, calculating speed as a function of time might have time constraints—imagine a rocket that takes off only after a countdown!