Write the domain of the function in interval notation. Write numbers as integers or simplified fractions. \[ f(x)=\frac{3 x-5}{2 x^{2}+11 x-6} \] Part: \( \mathbf{0} / 3 \) Part 1 of 3 \( f(x)=\frac{3 x-5}{2 x^{2}+11 x-6} \) will not be a real number when the (Choose one) \( \mathbf{~ i s ~ z e r o . ~} \)

To find the domain of the function \( f(x) = \frac{3x - 5}{2x^2 + 11x - 6} \), we need to identify when the denominator is equal to zero, since the function will be undefined at those points. First, we can factor the quadratic in the denominator: \[ 2x^2 + 11x - 6 = (2x - 1)(x + 6) \] Setting the factors equal to zero gives us the critical points: \[ 2x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{2} \] \[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \] Thus, the function \( f(x) \) will not be a real number at \( x = \frac{1}{2} \) and \( x = -6 \). Therefore, the domain in interval notation is: \[ (-\infty, -6) \cup (-6, \frac{1}{2}) \cup (\frac{1}{2}, \infty) \]