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In the world of mathematics, particularly in the study of functions, the concept of intersection points plays a pivotal role. These points are where two functions, when plotted on a graph, meet or cross each other. For example, when dealing with a piecewise function like the max function in our exercise, we're interested in identifying at what points the functions that compose the piecewise function have the same value.

To find these points, we set the functions equal to each other and solve for the variable. In this exercise, we were given functions like 1 - x, 1 + x, and the constant function 2. By setting each pair equal and solving, we obtained the solutions for x revealing the exact locations where our functions intersect.

Determining intersection points is crucial because it helps us to know where to separate our analysis of each function within the piecewise construction, ensuring a more precise understanding of the overall function's behavior.

The next logical step after finding the function intersection points is to define the intervals of interest. These intervals are segments on the x-axis that we divide based on the intersection points we've identified. They allow us to focus our investigation of the function's behavior on specific ranges where we expect certain conditions or rules to apply.

Consider our max function; once we determined the intersection points, we set up four distinct intervals. Within each interval, we can be sure that only one of the functions involved in the max function will dominate, meaning it will always have the highest value compared to the others in that range.

For instance, in Interval 1 (It's critical to select a representative value within each interval to properly assess which function will govern. These intervals and their evaluations are key to constructing the final piecewise representation of our maximum function, making them invaluable tools for analysis.

To fully understand a piecewise function like our max function, maximum function evaluation is an essential process. It involves picking a value within each of our previously defined intervals and evaluating all parts of our piecewise function. The highest resulting value will tell us which function is the 'maximum' function in that interval.

By performing these evaluations, as we did in the exercise, we clearly see which part of the max function to use based on the interval in question. This step is important because it directly influences the final form of our piecewise function, representing the maximum value across all intervals.

In our case, the evaluations of the three candidate functions 1 - x, 1 + x, and 2 within their respective intervals helped us conclude that for any value of x, the final piecewise function takes the form defined in Step 4. This formal process is what makes mathematical analysis powerful, offering a systematic approach for determining the behavior of complex functions.