91Ó°ÊÓ

An inequality is a mathematical statement that compares two expressions and shows that one expression is greater than, less than, or approximately equal to the other. In the context of this exercise, we deal with the inequality \(x - 2y + 4 \geq 0\). This type of inequality is common when determining the domain of functions involving square roots. The key here is to identify which values will make the expression under the square root non-negative since the square root of a negative number is undefined.

When solving an inequality, these are the general steps:

• Set the expression related to the inequality (like the one inside a square root) greater than or equal to zero.
• Re-arrange and simplify the expression to find values of unknown variables that satisfy the inequality.
• Interpret the solution in terms of "greater", "less", or "equal" to the relation that is established.

For any given function with an inequality, understanding and solving this allows us to determine where the function is defined and applicable.

The square root function is a mathematical operation that extracts the square root of a number or expression. In simple terms, it is the value that, when multiplied by itself, yields the original number. The square root symbol is represented as \(\sqrt{}\). For the function \(h(x, y)=\sqrt{x-2y+4}\), the expression inside the square root, \(x-2y+4\), must be non-negative (i.e., zero or greater).

This requirement arises because real numbers do not have real square roots, meaning the square root of a negative number does not exist within the real number system. Therefore, when dealing with square root functions, ensure:

• The expression under the square root remains non-negative.
• The inequality derived from the non-negative condition determines the domain of the function.

Using the concept of square root functions helps in identifying valid inputs for the function, ensuring all operations are mathematically sound.

Interval notation is a method used to represent a range of values, showing which numbers are included in a set. It offers a concise way to express the domain or solution set derived from inequalities.

In this exercise, after determining the inequality \(y \le \frac{1}{2}x+2\), we have expressed the domain in interval notation: \(\{(x,y) \in \mathbb{R}^2 \mid y \le \frac{1}{2}x+2\}\). This notation communicates that all ordered pairs \((x, y)\) in the set of real numbers are valid as long as \(y\) satisfies the condition of being less than or equal to a transformed \(x\)-based linear expression.

• "\(\{(x,y) \in \mathbb{R}^2 \mid ... \}\)" indicates that \((x,y)\) belongs to real numbers.
• The condition \(y \le \frac{1}{2}x+2\) is clearly stated within the set notation.

Using interval notation provides an efficient and clear method to specify solution sets, which is especially helpful in mathematical communications.