Here I will try to explain boolean expressions. Boolean expressions are expressions that evaluate to “true” or “false”. “True” is like a value of 1 and “false” is like a value of 0 (zero).

Suppose you have as your boolean for Y₁. If X is greater than , the expression can be replaced with a “1”. In turn, , so a value of true causes Y₁ to plot.

If X is equal to or less than , this same expression can be replaced with a 0. does not plot in that case. But since falls in the domain of Y₂, then it is Y₂ that will plot instead.

Suppose now I have two functions on my graphing calculator:

I have written the two booleans in such a way that only one of the statements can be true at a time. Notice that if the boolean for Y₁ us true, then the one for Y₂ is not. That means for X values less than , only Y₁ will plot. For all other values, only Y₂ will plot, because its boolean will now be true and Y₁‘s false. Ultimately, we obtain the graph:

This is produced by a piecewise function which can be expressed using the following standard notation:

Others have used different techniques for piecewise functions. One which seems to possess certain advantages would have been to place all details on to as the sum of both “pieces” where each “piece” is multiplied by the boolean of the restriction, as follows:

While being slightly unwieldy (being sure to run past the end of the display), it saves the user from having to flip between and using the key.

There was also a useful idea where you could disable plotting for Y₁ and Y₂; and, for a new function Y₃, enter: Y₁ + Y₂ using the VARS key to obtain the Y variables. The result is a single function that does not require flipping between functions with the or , and proving continuity is much easier. And, the function is more like a piecewise function rather than two separate functions.