Question

What is the domain of \(h(x, y)=\sqrt{x-y} ?\)

Short answer

Answer: The domain of the function is \(\lbrace (x, y) \in \mathbb{R}^2 ~|~ x \geq y \rbrace\).

Step-by-step solution

Write down the inequality condition for the function to be defined

The square root function requires a non-negative input. That means the argument inside the square root must be greater than or equal to zero: \(h(x, y) = \sqrt{x - y} \implies x - y \geq 0\)

Solve the inequality

To find the domain, we need to find the values of x and y that satisfy the inequality \(x - y \geq 0\): Adding y to both sides, we get: \(x \geq y\)

Express the domain

The domain consists of all possible (x, y) pairs that satisfy \(x \geq y.\) In set notation, the domain can be expressed as: Domain: \(\lbrace (x, y) \in \mathbb{R}^2 ~|~ x \geq y \rbrace\)

Key concepts

Square Root Function

Understanding the square root function is crucial when determining the domain of certain types of functions. The square root function, denoted as \(\sqrt{x}\), is defined only for x that is non-negative (\(x \geq 0\)). This is because the square root of a negative number is not a real number and hence, is not included in real number function definitions.

When dealing with a square root function such as \(h(x, y) = \sqrt{x - y}\), we are particularly interested in ensuring that the expression under the square root sign (\(x - y\)) is non-negative. This requirement directly influences the range of acceptable values for \(x\) and \(y\) which, in turn, determines the domain of the function. It's imperative for students to remember that the square root function opens upwards, starting from the origin (0,0) on a graph, and each output value is the positive root of the input value.

Inequality

Inequalities are mathematical statements used to show the relative size or order between two values. They are critical in determining the domain of a function where certain conditions must be met for the function to be defined. In our exercise, we work with the inequality \(x - y \geq 0\). Solving such an inequality helps us to find the set of all \(x\) and \(y\) values that make the function exist in the real number system.

When dealing with inequalities in the context of functions, it's essential to isolate the variable of interest. For instance, adding \(y\) to each side of \(x - y \geq 0\) gives us the simplified inequality \(x \geq y\). This expression tells us that for the square root function to be valid, the value of \(x\) must always be equal to or larger than \(y\). Recognizing how to manipulate and solve these inequalities plays a pivotal role in understanding the properties of the function in question.

Set Notation

Set notation is a standardized mathematical language used to describe collections of elements. It's particularly useful for expressing the domain or range of a function in a concise and clear way. For instance, the domain of \(h(x, y)\) from our exercise is represented in set notation as \(\lbrace (x, y) \in \mathbb{R}^2 ~|~ x \geq y \rbrace\), which translates to 'the set of all pairs \(x\), \(y\) in the plane of real numbers where \(x\) is greater than or equal to \(y\)'.

Breaking down this notation:

• \(\lbrace ... \rbrace\) denotes a set.
• \((x, y)\) represents ordered pairs of numbers.
• \(\mathbb{R}^2\) indicates the set of all such pairs in two-dimensional real number space.
• The vertical bar \(\|\) reads as 'such that' or 'where'.
• The inequality \(x \geq y\) is the condition elements of the set must satisfy.

Leveraging set notation helps in being precise about the conditions defining the domain and ensures a shared understanding when communicating mathematical concepts.