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The square root function is a crucial mathematical operation often encountered in algebra. It is represented by the symbol \( \sqrt{} \), indicating the principal square root of a number or expression. The principal aspect means we take the non-negative root, as every positive number technically has two square roots (one positive and one negative).

When you see a function involving a square root, such as \( f(s) = \sqrt{s - 2} + 5 \), it's important to remember a key property: the expression inside the square root must be greater than or equal to zero for the function to be defined. This is because you cannot take the square root of a negative number in standard real numbers. Therefore, any function involving a square root must consider this limitation.

• If \( s = 2 \), then \( \sqrt{s-2} = \sqrt{0} = 0 \).
• For any \( s > 2 \), \( s-2 \) is a positive number, and the square root is a real number.
• For \( s < 2 \), the expression inside the square root becomes negative, and in standard practices, the function is undefined.

Understanding this aspect helps in finding the domain, as only specific values for \( s \) will make the function computable.

Solving inequalities is akin to solving equations, but with a few more things to watch out for. In the case of the function \( f(s) = \sqrt{s - 2} + 5 \), to ensure the square root is defined, you need to solve the inequality \( s - 2 \geq 0 \).

This process involves determining what values of \( s \) satisfy the inequality. Simply rearrange the inequality to solve for \( s \):

• We start with the inequality \( s - 2 \geq 0 \).
• By adding 2 to both sides, you get \( s \geq 2 \).

The solution to this inequality tells us the range of values for which our original function is defined. Inequalities represent a range of values rather than a specific point, and solving them can sometimes also involve reversing inequality signs or handling different operations - but in this case, it is straightforward.

Interval notation is a concise way of expressing sets of numbers between specified lower and upper bounds. It is particularly useful when describing the domain of a function. For example, in the function \( f(s) = \sqrt{s - 2} + 5 \), we determined that the valid domain based on our inequality solution is \( s \geq 2 \).

Interval notation is used to express this domain as \([2, \, \infty)\), which is interpreted as follows:

• The square bracket \([\) indicates that 2 is included in the interval (closed interval), meaning \( s \) can be exactly 2.
• The parenthesis \()\) after infinity shows that infinity is not a precise number and is not included (open interval).

Understanding how to read and write interval notation effectively is essential in higher-level math to communicate the range of valid inputs concisely. It's a very efficient way of summarizing the continuous nature of solutions that inequalities often produce.