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Interval notation is a way of writing subsets of the real number line. It is particularly useful in expressing the domain or range of a function. In interval notation:

• Square brackets [ ] indicate that an endpoint is included in the interval.
• Parentheses ( ) mean that the endpoint is not included.
• A union ( ∪ ) combines multiple intervals into one set.

To represent all real numbers between two points, say \(a\) and \(b\), where \(a < b\), we write:
\begin{align*} (a, b) \text{ if } a < x < b \ [a, b] \text{ if } a \text{ and } b \text{ are included} \ (-\text{infinity}, a) \text{if it extends to negative infinity} \ [a, \text{infinity}) \text{if it extends to positive infinity} \ Now, let’s see how to use it in our example. In the final step, we get the domain of the function in the form of two distinct intervals: \ (-\text{infinity}, -\text{sqrt}(5)] \text{ and } [\text{sqrt}(5), \text{infinity}) \ Combining these, the domain in interval notation is written as: \ (-\text{infinity}, -\text{sqrt}(5)] \text{ U } [\text{sqrt}(5), \text{infinity}) \ This signifies that the function \(h(a) = \text{sqrt}(a^2 - 5)\) is defined for all values of \(a\) in these intervals.

A square root function is a function that involves the square root of an expression. These types of functions are written as \(f(x) = \text{sqrt}(expression)\). For the function to be defined, the expression under the square root must be non-negative because square roots of negative numbers are not real numbers.

In our example function \(h(a) = \text{sqrt}(a^2 - 5)\), the expression inside the square root is \(a^2 - 5\). To ensure the function is defined, \(a^2 - 5\) must be greater than or equal to zero: \ \begin{align*} a^2 - 5 \text{>= 0} \ a^2 \text{>= 5} \ This means that \(a\) can be either less than or equal to \(-\text{sqrt}(5)\) or greater than or equal to \(\text{sqrt}(5)\). Any number within these constraints will ensure that the value inside the square root is non-negative.

Quadratic inequalities involve expressions where the highest degree term is squared. They are usually in the form \(ax^2 + bx + c \text{>= 0}\), \(ax^2 + bx + c \text{<= 0}\), \(ax^2 + bx + c > 0\), or \(ax^2 + bx + c < 0\). Solving them involves finding the values of the variable that satisfy the inequality.

Steps to solve quadratic inequalities:

• Set the inequality to zero (just like in solving quadratic equations).
• Factorize the quadratic expression if possible.
• Determine the critical points where the expression equals zero.
• Use test points to determine which intervals satisfy the inequality.

In the example of \(h(a) = \text{sqrt}(a^2 - 5)\), we solve the inequality \(a^2 - 5 >= 0\) by: \ \begin{align*} a^2 \text{>= 5} \ This results in two intervals where \(a\) can lie: \ \text{a <= -sqrt}(5) \text{ or } \text{a >= sqrt}(5) \$ Combining these intervals gives us the complete solution in terms of the domain: \ (-\text{infinity}, -\text{sqrt}(5)] \text{ U } [\text{sqrt}(5), \text{infinity})