Domain and Range of Function f(x) = 2 - |x - 1|

Domain:
The domain of a function is the set of all possible values of x for which the function is defined. In this case, there are no restrictions on the value of x that can be plugged into the function. Therefore, the domain of f(x) = 2 - |x - 1| is all real numbers.

Range:
The range of a function is the set of all possible values of y that can be obtained by plugging in the values of x in the function. To determine the range of this function, we need to find the minimum value of f(x).

When x < 1,="" the="" expression="" (x="" -="" 1)="" is="" negative.="" therefore,="" |x="" -="" 1|="-(x" -="" 1)="1" -="" x.="" substituting="" this="" value,="" we="" get="" f(x)="2" -="" (1="" -="" x)="x" +="" />

When x >= 1, the expression (x - 1) is non-negative. Therefore, |x - 1| = (x - 1). Substituting this value, we get f(x) = 2 - (x - 1) = 3 - x.

Thus, the function f(x) takes on values of x + 1 when x < 1="" and="" 3="" -="" x="" when="" x="" />= 1. The minimum value of x + 1 is 0, which occurs when x = -1. The maximum value of 3 - x is 2, which occurs when x = 1. Therefore, the range of f(x) is (-infinity, 2].

Final Answer:
Thus, the domain of f(x) = 2 - |x - 1| is all real numbers and the range is (-infinity, 2]. Therefore, the correct option is (B).