Final answer:
The domain of the rational expression (4x-16)/(x(x-4)) is all real numbers except x = 0 and x = 4, because the expression is undefined when the denominator equals zero.
Step-by-step explanation:
Finding the Domain of a Rational Expression
To find the domain of the rational expression (4x-16)/(x(x-4)), we need to identify the values of x for which the expression is undefined. The expression is undefined when the denominator equals zero since division by zero is not allowed in mathematics. Thus, we must solve the equation x(x-4) = 0 to find the values that are not in the domain.
Setting the denominator equal to zero gives us two equations to solve:
x = 0, and
x - 4 = 0, which leads to x = 4.
The solutions x = 0 and x = 4 are the values for which the denominator becomes zero, so they must be excluded from the domain. Therefore, the domain of the rational expression is all real numbers except x = 0 and x = 4.