Math  Help

Domain and Range

What is the domain and range?

Domain is a set of all input values of a function whereas range considers the set of all the output values of a function.

On a coordinate plane, domain looks at x- values (horizontal axis) of a graph whereas range is the set of y-values (vertical axis) of a given function.

Examples of domain and range

1. Determining domain and range, given a table.

The domain is the set: {-3,-2,-2,0,1}

The range is the set: {7,2,-1,-2}. There are 4 terms in the range since -1 repeats itself twice.

2. Determining domain and range given mapping diagrams.

The domain is: {a,c}. b is not included in the domain as the input results in no output.

The range is: {1,3}. 2 and 4 are not included in the range as no letter in the input results in that output.

Note: Since domain and range is a set of numbers, braces { } are used to denote the set of numbers.

3. Determining the domain and range of continuous functions.

Let’s take a look at the graph. Notice the two arrows at each end.

Here, the graph touches every x value possible and every y-value due to the arrows pointing in both directions.

Hence, the domain is:

Range is:

Let us look at another graph. Notice the graph has a starting point and one arrow. This means it is only continuous at one end.

The domain is: since all the x-values the graph touches are greater than 1

The range is: since all the y-values, the graph with touch are greater than 2

Next, look at a graph that has no arrows pointing which means it starts at one point and ends at another point.

The smallest x-value is -1 and the largest x value this graph will touch is 1 which means all the possible values of x are between -1 and 1.

Hence, the domain is:

Similarly, the range is:

4. Determining the domain and range of open and closed circles

An open circle means the graph does not include that point and is called a hole. A closed circle means it does include the point.

The domain of the function is: { x∈R | -3 ≤ x < 3 }. Here, since the graph never touches x = 3 due to the open circle eventhough it comes very close to x = 3 (so 2.9999) hence we say x is less than 3, just not equal to 3.

The range of the function is: { y∈R | -4 ≤ y < -2 }. For the range, even though there is a hole at y=-2, it can be filled by the closed circle also at y=-2. Hence, y values of the function are greater than y=-4 and less than y=-2.

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