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A square root function involves finding the root of a value, meaning you are looking for a number that multiplies by itself to give the original value. In mathematical notation, the square root of a number is represented with the radical symbol \( \sqrt{} \). For any square root function to be defined, the value under the square root must be non-negative (i.e., zero or positive).

For example, consider the expression \( \sqrt{3 - t} \). Here, \( 3 - t \) must be greater than or equal to zero so that the square root expression remains defined. This leads to the inequality \( 3 - t \geq 0 \), which simplifies to \( t \leq 3 \). As a general rule, when dealing with any square root function, always ensure the inside of the square root is zero or positive. That is because the square root of a negative number is not real in standard real number arithmetic.

Inequality solving involves finding the set of values that satisfy a given inequality. An inequality might look like \( 3 - t \geq 0 \), which means we need to find all the values of \( t \) that make the expression true.

To solve \( 3 - t \geq 0 \), you rearrange the inequality to isolate \( t \) on one side. Subtract \( t \) from both sides to get \( 3 \geq t \). This can also be written as \( t \leq 3 \).

When solving inequalities, always pay attention to the inequality symbols:

• \( \geq \) means "greater than or equal to". The solution includes the boundary value.
• \( \leq \) means "less than or equal to".
• Reversing the direction of an inequality is necessary when multiplying or dividing by a negative number.

Inequality solving plays a critical role in identifying valid inputs for functions with conditions like non-negative squares roots, ensuring all expressions are valid within their defined domains.

Interval notation is a mathematical way to describe sets of numbers, typically representing all numbers between two endpoints. It's especially useful for expressing the domain of a function.

For the function \( g(t) = \sqrt{3-t} - \sqrt{2+t} \), we found the domain to be between \( -2 \) and \( 3 \) inclusive, combining the inequalities \( 3 - t \geq 0 \) and \( 2 + t \geq 0 \).

To write this domain in interval notation:

• Use square brackets \([ \)] to indicate that the endpoints are included in the interval.
• The interval starts at the smallest number, \(-2\), and ends at the largest number, \(3\).
• Thus, the domain \([-2, 3]\) includes all numbers from \(-2\) to \(3\), including the endpoints.

This notation is concise and useful for conveying ranges of values quickly and clearly, indicating exactly where a function is defined and valid for use.