Evaluate the piecewise function at the specified intervals and provide the solution in JSON format.

Solution:

The problem asks to analyze a piecewise function defined as:

• $$y = x + 2$$, if $$-4 \le x \le -1$$
• $$y = x^2$$, if $$-1 \le x \le 2$$
• $$y = 4$$, if $$2 < x \le 5$$

This is a mathematical problem involving a piecewise function. To provide a full analysis, one would typically:

• Evaluate the function at specific points within each interval.
• Determine the behavior of the function in each interval (e.g., linear, quadratic, constant).
• Identify any points of discontinuity or continuity.
• Potentially sketch the graph of the function.

Since no specific value of 'x' or interval for evaluation is provided in the prompt, and the request is to 'evaluate' without further context, a general analysis of the function's definition is presented.

Analysis of intervals:

• Interval 1: $$-4 \le x \le -1$$. The function is linear: $$y = x + 2$$. This segment starts at $$y = -4 + 2 = -2$$ (at $$x = -4$$) and ends at $$y = -1 + 2 = 1$$ (at $$x = -1$$).
• Interval 2: $$-1 \le x \le 2$$. The function is quadratic: $$y = x^2$$. This segment starts at $$y = (-1)^2 = 1$$ (at $$x = -1$$) and ends at $$y = 2^2 = 4$$ (at $$x = 2$$). Note that at $$x=-1$$, the value from this interval ($$1$$) matches the value from the first interval, indicating continuity at this point.
• Interval 3: $$2 < x \le 5$$. The function is constant: $$y = 4$$. This segment starts just after $$x = 2$$ (where $$y=4$$) and ends at $$y = 4$$ (at $$x = 5$$). At $$x=2$$, the value from the second interval is $$4$$, matching the constant value of the third interval, indicating continuity at this point as well.