How to Apply Multiple Constraints in GeoGebra

In GeoGebra, it is often useful to constrain an object or a function to a specific range of values, or to simultaneously constrain multiple variables.

To do this, you can combine conditions using the logical AND operator, written either as ∧ or as &&.

condition1 && condition2

In addition to the AND operator (&&), you can also combine conditions with the logical OR operator ||, and apply negation using the logical NOT operator !.

For example, suppose you want to display the function $f(x, y) = x^2 + y^2$ only when both of the following conditions are satisfied:

• $ -1 ≤ x ≤ 1 $
• $ -1 ≤ y ≤ 1 $

In GeoGebra, you can achieve this by using the If command together with the and operator.

Type the following command and press Enter:

If[-1 ≤ x ≤ 1 && -1 ≤ y ≤ 1, x^2 + y^2, NaN]

As a result, GeoGebra will plot the function $x^2 + y^2$ only for values of $x$ and $y$ within the specified range. Outside this range, nothing will be displayed ("NaN" stands for Not a Number).

If instead you want to visualize the valid region in the plane directly (for example, a rectangle representing the constrained range), you can simply enter the combined inequalities:

-1 ≤ x ≤ 1 && -1 ≤ y ≤ 1

GeoGebra will then display the region of points $(x, y)$ that satisfy both conditions - in this case, a square centered at the origin.