Final answer:
The domain of f(x) = x/(x^2 - 2x) is all real numbers except 0 and 2, and the range is all real numbers except 0.
Step-by-step explanation:
The function in question is f(x) = x/(x^2 - 2x). To find the domain of f(x), we must identify all the values of x for which the function is defined. Looking at the denominator of the function (x^2 - 2x), it is clear that we cannot have values that make the denominator zero, as division by zero is undefined.
Factoring the denominator, we get x(x - 2).
Therefore, x cannot be 0 or 2. So, the domain of f(x) is all real numbers except x = 0 and x = 2.
For the range of f(x), we must look at all possible values that f(x) can take.
As x approaches the values that are excluded from the domain (0 and 2), f(x) will tend towards infinity or negative infinity, indicating that all real numbers can potentially be in the range.
However, since no x-value will make f(x) equal to zero, the only value not in the range is 0.
Thus, the range of f(x) is all real numbers except 0.