Determine where the following function is continuous. \( M(x)=\frac{x-3}{9 x^{2}-64} \) The function is continuous on \( \square \). (Type your answer in interval notation. Use inlegers or fractions for any numbers in the expression.)

To determine where the function \( M(x) = \frac{x-3}{9x^{2}-64} \) is continuous, we need to identify where the denominator is not zero, since division by zero is undefined. 1. **Find the zeros of the denominator:** \[ 9x^2 - 64 = 0 \\ 9x^2 = 64 \\ x^2 = \frac{64}{9} \\ x = \pm \frac{8}{3} \] So, the denominator is zero at \( x = \frac{8}{3} \) and \( x = -\frac{8}{3} \). 2. **Determine the continuity:** The function \( M(x) \) is a rational function and is continuous everywhere except where the denominator is zero. Therefore, \( M(x) \) is continuous for all real numbers except \( x = \frac{8}{3} \) and \( x = -\frac{8}{3} \). 3. **Express the continuity in interval notation:** \[ (-\infty, -\frac{8}{3}) \cup (-\frac{8}{3}, \frac{8}{3}) \cup (\frac{8}{3}, \infty) \] **Answer:** \((-\infty, -\frac{8}{3}) \cup (-\frac{8}{3}, \frac{8}{3}) \cup (\frac{8}{3}, \infty)\)