Which bracket means inclusive

That notation looks like this:. Well, let us get just a bit more complicated. Using interval notation we will show the set of number that includes all real numbers except 5. First, stated as inequalities this group looks like this:. The statement using the inequalities above joined by the word or means that x is a number in the set we just described, and that you will find that number somewhere less than 5 or somewhere greater than 5 on the number line.

In interval notation a logically equivalent statement does not use the word or, but rather a symbol for what is called the union of two groups of numbers. The symbol for union coincidentally looks like a U, the first letter of union. However, it is really not a letter of the alphabet. Here is what the union symbol looks like:. So, the group of numbers that includes all values less than 5 and all values greater than 5, but does not include 5 itself, expressed as interval notation looks like this:.

Let us consider one last set of numbers. What is the domain and range for 3,1 , 1,-4 , and 2, 8? What is the domain and range of a linear function?

Is domain the independent or dependent variable? How do you find the domain and range of a function in interval notation? How do you find domain and range of a rational function? How do you find domain and range of a quadratic function? Alternative mnemonic, if you put the square brackets back to back in reverse order it looks like a capital I for Inclusive: ][.

Or if you put parenthesis back to back in reverse order it looks like an X for eXclusive: — Brad Cupit. Show 2 more comments. Active Oldest Votes. Improve this answer. Stefan 1 1 gold badge 3 3 silver badges 16 16 bronze badges. Michael Mrozek Michael Mrozek k 25 25 gold badges silver badges bronze badges.

But it's a valid cardinal number when answering questions like "How many integers are there? It's also, as in this case, perfectly valid as a limit — Kevin Wright.

Show 3 more comments. A closed interval [a,b] includes the end points. An open interval a,b excludes them. Mark Byers Mark Byers k gold badges silver badges bronze badges.

Add a comment. The brackets [ and ] means: The number is included , This side of the interval is closed , The parenthesis and means: The number is excluded , This side of the interval is open. An interval with mixed states is called "half-open". Array indexes tend to use a different offset depending on which field are you in: Mathematics tends to be one -based.

Integers If we have a set or array, say of the first few primes [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ] , Mathematicians would refer to the first element as the 1st absolute element. Community Bot 1 1 1 silver badge. Michaelangel Michaelangel 2, 1 1 gold badge 22 22 silver badges 20 20 bronze badges. Eddy Eddy 1, 2 2 gold badges 17 17 silver badges 32 32 bronze badges.

The range is the set of possible output values, which are shown on the y -axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values.

Figure Energy Information Administration. In interval notation, the domain is [, ], and the range is about [, ]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Given Figure , identify the domain and range using interval notation. For example, the domain and range of the cube root function are both the set of all real numbers. We will now return to our set of toolkit functions to determine the domain and range of each.

Both the domain and range are the set of all real numbers. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers. The same applies to the vertical extent of the graph, so the domain and range include all real numbers. Further, 1 divided by any value can never be 0, so the range also will not include 0.

Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative it is an odd function.

Given the formula for a function, determine the domain and range. There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result. We cannot take the square root of a negative number, so the value inside the radical must be nonnegative. We then find the range. Sometimes, we come across a function that requires more than one formula in order to obtain the given output. It is the distance from 0 on the number line.

All of these definitions require the output to be greater than or equal to 0. Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain. Tax brackets are another real-world example of piecewise functions.

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:. Given a piecewise function, write the formula and identify the domain for each interval.

Two different formulas will be needed. The function is represented in Figure. Find the cost of using 1. To find the cost of using 1. Because 1. Each of the component functions is from our library of toolkit functions, so we know their shapes.

We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality. Figure shows the three components of the piecewise function graphed on separate coordinate systems.

Now that we have sketched each piece individually, we combine them in the same coordinate plane. Can more than one formula from a piecewise function be applied to a value in the domain? Each value corresponds to one equation in a piecewise formula. Access these online resources for additional instruction and practice with domain and range. The domain of a function depends upon what values of the independent variable make the function undefined or imaginary. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

Graph each formula of the piecewise function over its corresponding domain. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. For the following exercises, find the domain of each function using interval notation.