In Exercises 31–50, find f/g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Function division involves creating a new function by dividing one function by another. In this case, f/g means taking the function f(x) = x - 5 and dividing it by g(x) = 3x². The resulting function will be expressed as (x - 5) / (3x²), which is essential for further analysis.

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f/g, we must identify values of x that do not make the denominator zero, as division by zero is undefined. Thus, we need to solve the equation 3x² = 0 to find any restrictions on the domain.

Finding restrictions involves determining the values that must be excluded from the domain of a function. In this case, we set the denominator g(x) = 3x² equal to zero to find x = 0. Therefore, the domain of the function f/g excludes x = 0, leading to the domain being all real numbers except zero.