In this problem, we're asked to graph the following piecewise function. To graph this function, we first need to identify the boundaries on our graph. For the first part of the function, we note that x is less than negative 2, so we draw a line at x equals negative 2 to represent this boundary.
Next, the function specifies that x is between negative 2 and 1. Thus, we draw another line at x equals 1. The function for values of x greater than or equal to 1 is represented to the right of this line. As such, the function is divided into three distinct parts between these boundaries.
Starting with the first segment of the equation, it states that the function is equal to negative 4, a constant value. Graphically, this is represented as a horizontal line at y equals negative 4, extending indefinitely to the left but stopping at x equals negative 2.
Moving on to the second part, the function is described by x plus 1. This linear function intersects the y-axis at 1 and continues to increase. This line will only exist between x equals negative 2 and x equals 1, constrained by the defined boundaries.
For the final part, the function is x squared, which forms a parabola. This part of the function starts at x equals 1, where the value of the function is 1 (since 1 squared equals 1), and extends indefinitely to the right, increasing in value.
When considering where to place dots or open circles to indicate boundaries of definition, observe the inequalities defining each segment. For the first segment (x less than negative 2), we place an open circle at x equals negative 2. For the line segment defined from negative 2 to 1, we place a solid dot at x equals negative 2 and an open circle at x equals 1 because x is strictly less than 1. For the final segment starting at x equals 1, a solid dot at this point indicates that x equals 1 is included in the domain of this part of the function.
This visualization helps to understand how the piecewise function is structured and ensures clarity in its graphical representation. Such practice is essential for mastering the graphing of complex functions.