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The range of a function is the set of all possible output values it can produce. For any given function, identifying the range tells us what kind of values we can expect from it. In the context of the function \( f(x, y) = \sqrt{2 + x - y} \), determining the range involves understanding the behavior of the square root function.

Since a square root outputs only non-negative values (i.e., zero or positive numbers), the range of \( f(x, y) \) is limited to non-negative real numbers. This is true because the expression under the square root, \( 2 + x - y \), ensures that the resulting square root is always zero or more, provided \( y \leq x + 2 \). Thus, regardless of the input values \((x, y)\) that satisfy this inequality, the output of \( f(x, y) \) will always be non-negative, forming its range.

The square root function is a type of function represented in the form \( \sqrt{x} \). It is important because it only produces non-negative outputs. This characteristic is because the square of a real number is never negative, making the square root of any non-negative number defined and non-negative. Square roots can introduce specific constraints on the input values, as they are only real and defined for non-negative numbers.

In the given function \( f(x, y) = \sqrt{2 + x - y} \), the expression within the square root, \(2 + x - y\), must be non-negative to keep the square root defined over the real numbers. Negative results under the square root would lead to \(\sqrt{\text{negative number}}\), which are not considered real numbers. Consequently, the expression \(2 + x - y\) provides an important condition for the defined inputs or the domain of the function.

Inequalities are mathematical expressions that depict the relationship between different values, indicating whether one value is greater than, less than, or equal to another. They are crucial in determining domains and ranges in functions, particularly when working with functions that have specific operational limits.

In our example, the inequality \(2 + x - y \geq 0\), which simplifies to \(y \leq x + 2\), is critical in identifying valid input pairs \((x, y)\). This is because only pairs where this inequality holds will result in a non-negative value under the square root, hence producing real-valued outputs.

Understanding and solving inequalities helps define the domain (all the acceptable inputs) and implicitly the range (all the possible outputs) of the function, ensuring that the function outputs valuable and valid results that adhere to its mathematical constraints.