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Square root functions are mathematical expressions where the variable x is inside a square root. For the function to be valid, the value under the square root must be non-negative (i.e., zero or positive).
This is because the square root of negative numbers is not defined within the real number system. In simpler terms, you can't take the square root of a negative number and get a real number as a result.
Hence, when dealing with square root functions, always ensure the input makes the content inside the root zero or more. This is crucial for determining the domain of the function.

• Example: For the function \(f(x) = \sqrt{x-3}\), the expression under the square root is \(x-3\).
• For \(f(x)\) to be defined, \(x-3\) must be greater than or equal to zero.

Understanding these basics helps in identifying where a square root function can "live" on the number line.

Inequalities are expressions that show the relationship between two values where they may not necessarily be equal. In mathematical terms, inequalities tell us which quantity is less than, greater than, less than or equal to, or greater than or equal to another quantity.
In the context of determining domains, inequalities are essential because they help define the range of valid inputs for a function.

• If you have an inequality like \(x - 3 \geq 0\), it tells you that \(x\) must be equal to or larger than 3 to satisfy the inequality.
• Solving the inequality involves isolating the variable on one side to figure out the acceptable values it can take.

For our exercise, solving \(x - 3 \geq 0\) involves a straightforward step: adding 3 to both sides to get \(x \geq 3\). This further aids in expressing our mathematical findings in interval notation.

Interval notation is a concise way to represent a range of values, especially when dealing with domains and solutions of inequalities. It uses brackets and parentheses to show inclusivity or exclusivity of endpoints.

• Square brackets [ ] are used when an endpoint is included in the interval. This means the value is part of the solution.
• Round parentheses ( ) are used when an endpoint is not included in the interval.

For example, for the function \(f(x)=\sqrt{x-3}\), after solving the inequality \(x \geq 3\), we find the function is defined for all \(x\) values starting at 3 and going towards positive infinity.
In interval notation, this is written as \([3, +\infty)\), indicating that 3 is included in the domain, but infinity is not a limit we can reach. This easy-to-read form quickly communicates the domain of any mathematical function.
Understanding this notation is vital for sharing mathematical information effectively.