Final answer:
The overall range of the piecewise function f(x) = { -2x if x ≤ 0; 2x+1 if x > 0 } is from 0 to positive infinity, inclusive of the value 1.
Step-by-step explanation:
The question asks us to determine the range of a piecewise function given by f(x) = { -2x if x ≤ 0; 2x+1 if x > 0 }. To find the range, we need to consider the possible values that f(x) can take based on the two different rules for x values less than or equal to zero, and for x values greater than zero.
For the portion of the function where x ≤ 0, we have f(x) = -2x. Since x is non-positive, multiplying by -2 makes f(x) non-negative, and as x decreases, f(x) increases without bound. Therefore, the range for this part extends from 0 to positive infinity.
In the case where x > 0, the function is f(x) = 2x + 1. Here, as x increases, f(x) also increases, starting from a minimum value of 1 when x is just greater than 0. Thus, the range for this section also extends from 1 to positive infinity.
Combining these results, the overall range of the function is from 0 to positive infinity, and it includes the value 1 since for any epsilon > 0, there exists an x such that 0 < x < epsilon and f(x) = 2x + 1 will be in the interval (0,2).