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A square root function generally takes the form \(f(x) = \sqrt{x} \). This is a type of radical function characterized by the presence of a square root. These functions are only defined for values that make the expression inside the square root non-negative because the square root of a negative number is not a real number.

In the case of the square root function \(f(x) = \sqrt{x+2} \), you're tasked with determining where the square root is defined. Since square roots require non-negative inputs, the condition is that \(x+2\) should be greater than or equal to zero. This plays into determining the domain of the function, ensuring every input results in a real number output.

Solving inequalities is crucial when determining where certain functions are valid, like square root functions. Solving an inequality involves finding all values of the variable that make the inequality true.

Let's take the inequality \(x+2 \geq 0\). Here, you want to find all \(x\) values that satisfy this condition. To simplify, subtract 2 from both sides to isolate \(x\). This process will yield \(x \geq -2\). This means every number greater than or equal to \(-2\) will satisfy the inequality.

After solving the inequality, you have a clearer picture of where the function is defined because it shows the permissible values for \(x\). This is critical for determining the domain of functions involving square roots.

Interval notation provides a way to express a range of values succinctly. It's often used to indicate the domain or range of a function. When writing these intervals, you'll use brackets and parentheses.

For the domain of \(f(x) = \sqrt{x+2}\), once you solve the inequality \(x \geq -2\), you can express the domain using interval notation. You write the interval notation as \([-2, \infty)\), where:

• The square bracket \([-2,\) indicates that \(-2\) is included in the interval, meaning \(x\) can be equal to \(-2\).
• The parenthesis \(\infty)\) denotes that infinity is not a number that can be reached; hence it is not included. It indicates that there is no upper limit.

Interval notation provides a neat and standardized way to describe domains and is especially useful in calculus and higher mathematics where precision is crucial.