Determine the domain of the function: $y=\frac{3 x+2}{x^{2}-3 x+2}$

Step 1 ：We are given the function \(y=\frac{3x+2}{x^{2}-3x+2}\).

Step 2 ：The domain of a function is the set of all possible input values (often denoted as x) which will produce a valid output.

Step 3 ：In this case, the function is a rational function, which means it is a ratio of two polynomials.

Step 4 ：The only values that are not in the domain of a rational function are those that make the denominator equal to zero, because division by zero is undefined.

Step 5 ：Therefore, to find the domain of this function, we need to find the values of x that make the denominator equal to zero.

Step 6 ：The denominator of the function is \(x^{2}-3x+2\).

Step 7 ：The solutions to the equation \(x^{2}-3x+2=0\) are \(x=1\) and \(x=2\).

Step 8 ：These are the values that make the denominator of the function equal to zero, and therefore, they are the values that are not in the domain of the function.

Step 9 ：\(\boxed{\text{The domain of the function } y=\frac{3x+2}{x^{2}-3x+2} \text{ is all real numbers except } x=1 \text{ and } x=2. \text{ In interval notation, this can be written as } (-\infty, 1) \cup (1, 2) \cup (2, \infty).}\)