When you're learning about functions in Algebra II, it’s important to know about open and closed intervals. These help you understand what values a function can use (domain) and what values it can give back (range).

Open Intervals:

An open interval is shown as $(a, b)$. It includes all the numbers between $a$ and $b$, but not the endpoints $a$ and $b$.

For example, let’s look at the function $f(x) = \sqrt{x}$. Its domain can be written as $[0, \infty)$. This means $0$ is included (you can use it in the function), but there’s no upper limit, so it goes on endlessly.

Closed Intervals:

Closed intervals are shown like this: $[a, b]$. They include the endpoints $a$ and $b$.

Take the function $f(x) = 1/(x - 2)$. Its domain is $(-\infty, 2) \cup (2, +\infty)$. This means $x=2$ is not included because the function doesn’t work at that point.

Defining Domain and Range:

Choosing between open and closed intervals helps us figure out where a function can work (domain) and what values it can produce (range).

For example, the function $g(x)=x^2$ can take any real number. So, its domain is $(-\infty, \infty)$. However, the outputs are always $0$ or higher, so its range is $[0, \infty)$.

In short, knowing if intervals are open or closed helps us understand what inputs and outputs a function can have. This makes it easier to see how the function behaves!