91Ó°ÊÓ

The square root function, symbolized as \(y = \sqrt{x}\), is a fundamental concept in mathematics. It only produces non-negative outputs since you cannot take the square root of a negative number without involving complex numbers. In the given equation \(y = \sqrt{x-4}\), the expression under the square root must be non-negative. The number inside the square root function is called the *radicand*. For example, in \(\sqrt{(x-4)}\), \((x-4)\) is the radicand. This radicand determines the domain of the function. Since the square root of a number must be zero or positive, the radicand \(x - 4\) must be greater than or equal to zero. Consequently, this places restrictions on the input (the domain) to ensure valid outputs (the range).

Inequalities are mathematical statements that show the relationship between values, stating whether one is bigger, smaller, or equal to another. They use symbols such as:

• \(<\) less than
• \(>\) greater than
• \(\leq\) less than or equal to
• \(\geq\) greater than or equal to

In our case, we used an inequality to determine the domain of \(y = \sqrt{x-4}\). We set \(x-4 \geq 0\) because the expression inside the square root must be non-negative. By solving this inequality, we find \(x \geq 4\). This tells us that x must be 4 or larger for the given function to be valid.

Interval notation is a way to describe the set of numbers that form a part of the domain or range of a function. It uses brackets to show which numbers are included in the set:

• \([a, b]\) means all numbers from a to b, including both a and b
• \((a, b)\) means all numbers from a to b, but not including a and b
• \([a, b)\) means all numbers from a to b, including a but not b
• \((a, b]\) means all numbers from a to b, including b but not a

In our scenario, the domain of \(y = \sqrt{x-4}\) is \([4, \infty)\). This indicates that the function accepts any x-value starting from 4 (included) and extending to positive infinity. The range in this case is \([0, \infty)\), meaning the output y can start from 0 (included) and go up indefinitely as x increases.