Function Notation

Function notation is a way to denote the relationship between two sets of numbers, typically the input and output of a function. The notation expresses the idea that for every value of $x$, there is a corresponding function value $f(x)$, which is the output.

It is important to note that a composite function can exist. This means the two functions for instance, $f(x)$ and $g(x)$ are combined, which is denoted as $g(f(x))$. To solve this, the value of a certain value of $f(x)$, must further be substituted into the equation given by $g(x)$.

Bracket Interval Notation

Interval notation is used to describe sets of numbers, typically indicating the domain or range of a function. As such, the notation is often able to specify the intervals where a function is increasing, decreasing, or constant. Intervals can be open, closed, or a combination of both, and are represented using parentheses; ( ) for open intervals and square brackets [ ] for closed intervals.

Open intervals written in form $(a,b)$ indicate that all numbers between $a$ and $b$ are included. Closed intervals denoted as $[a,b]$, indicate that all numbers that lie between $a$ and $b$ are including, while $a$ and $b$ are excluded from the equation.

This can be represented on a diagram. For example, the inequality $1\leq x\leq5$, can be written in interval notation as $[1,5]$, which is shown on a diagram below:

Now, if we were to change the interval slightly so that it becomes $[1,5)$, the diagram changes.

As we can see, the dot located above 5 is not shaded in. This is how to represent a number that is not included in the number range.