# Boolean expressions on the TI-84 for Piecewise functions

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 $(x>-1.0881)$ as your boolean for Y1. If X is greater than $-1.0881$, the expression can be replaced with a “1”. In turn, $1 \times (1.5x + 3) = 1.5x + 3$, so a value of true causes Y1 to plot.

If X is equal to or less than $-1.0881$, this same expression can be replaced with a 0. $Y_1$ does not plot in that case. But since $X \leq -1.088$ falls in the domain of Y2, then it is Y2 that will plot instead.

Suppose now I have two functions on my graphing calculator:

$Y_1 = (1.5x + 3)(x < -1.0881)$
$Y_2 = (2x^2 - 1)(x \geq -1.0881)$

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 $-1.0881$, 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 $Y_1$ as the sum of both “pieces” where each “piece” is multiplied by the boolean of the restriction, as follows:

$Y_1 = (1.5x + 3)(x < -1.0881) + (2x^2 - 1)(x \geq -1.0881)$

While being slightly unwieldy (being sure to run past the end of the display), it saves the user from having to flip between $Y_1$ and $Y_2$ using the $\uparrow$ 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 $\uparrow$ or $\downarrow$, and proving continuity is much easier. And, the function is more like a piecewise function rather than two separate functions.