Final answer:
To find the points of discontinuity for the piecewise function, limits of the function segments at x = -2 and x = 2 should be examined. If the limits don't match when approaching from both sides, or the function does not approach a real value, the function is discontinuous at that point.
Step-by-step explanation:
To locate the points of discontinuity of the given piecewise function, we need to examine the behavior at the points where the function changes its form, specifically at x = -2 and x = 2.
For the first segment of the function f(x) = 2ˣ⁺ - 1, we need to consider the limit as x approaches -2 from the left.
For the second segment f(x) = √x+2, we need to consider the limit as x approaches -2 from the right, and similarly for the limit as x approaches 2 from the left.
The third segment f(x) = 1/4x + 3/2, requires us to look at the limit as x approaches 2 from the right.
If the limits on either side of these points are not equal or the function does not approach real values, the function will have discontinuities at these points. In this specific case, the piecewise function changes at x = -2 and x = 2, suggesting that we analyze the values of each piece at these endpoints to determine continuity. If the pieces do not connect seamlessly, these are the points of discontinuity.