Determine the domain of: $f(x)=\frac{3 x}{(x-2)(x+7)}$

Step 1 ：The function given is \(f(x)=\frac{3 x}{(x-2)(x+7)}\).

Step 2 ：The domain of a function is the set of all possible input values (x-values) which will output real numbers.

Step 3 ：In this case, the function is a rational function. The denominator of a rational function cannot be zero because division by zero is undefined in mathematics.

Step 4 ：Therefore, to find the domain of this function, we need to find the values of x that make the denominator zero and exclude them from the domain.

Step 5 ：The denominator of the function is \((x - 2)*(x + 7)\).

Step 6 ：The values that make the denominator zero are -7 and 2. Therefore, these values should be excluded from the domain of the function.

Step 7 ：\(\boxed{\text{The domain of the function } f(x)=\frac{3 x}{(x-2)(x+7)} \text{ is } x \in \mathbb{R} \setminus \{-7, 2\}, \text{ which means all real numbers except -7 and 2. In interval notation, this can be written as } (-\infty, -7) \cup (-7, 2) \cup (2, \infty)}\)