# Domain And Range Calculator

Using the Domain and Range Calculator, you can check a function’s range and domain online for free. Studyqueries’s online calculator for domains and ranges makes the process faster, and the result is displayed in a fraction of a second.

## How to Use the Domain and Range Calculator?

The domain and range calculator can be used by following the steps below:

• Step 1: Enter the function into the input field
• Step 2: Click the “Calculate Domain and Range” button to see the output
• Step 3: The new window will display the domain and range

## What Is the Domain and Range of a Function?

An input domain, or domain of a function, is a set of values that a function can be used to evaluate. This domain is represented by the oval on the left in the image below. A value is provided by the function, f(x), for every member of the domain. As shown in the illustration below, the range of the function is the set of values it outputs, and these values are indicated by the right-hand oval. Functions are relationships that take inputs in one domain and output values in another. The rule for a function is that every input will yield exactly one output.

### Mapping of a Function

The oval on the left represents the domain of the function f, and the oval on the right represents the range. The green arrows show how each member of the domain is mapped to a particular value in the range.

The illustration shows that every value of the domain has a green arrow pointing to the corresponding value of the range. Thus, this mapping is a function.

We can also see this is a function by the list of ordered pairs since none of the x-values repeat: (*1,1), (1,1), (7,49), and (0.5,0.25) because each input maps to exactly one output. (It should be noted that although the output value of 11 repeats, only the input value cannot)

This mapping and set of ordered pairs are also indicative of a function based on the graph of the ordered pairs because the points do not form a vertical line. In the case of an x value repeating, there would be two points, which would not equate to a function. Take a look at this mapping and list of ordered pairs graphed on a Cartesian plane.

It is important to remember that not all functions have real numbers as their domain. As an example, the function is not defined when x=0 because you cannot divide a number by 0. If f represents a real number, then its domain is the set of all real numbers except 0. This is x≠0. Therefore, the domain of this function is R-{0}.

### Visualizing Domain and Range

The domain values are mapped to values in the range, which are visualized as graphs of functions

Key Points

• The values in the domain map onto the values in the range.
• It is possible to determine what type of relationship exists between the domain and range by using horizontal and vertical line tests.

Key Terms

• Range: The set of values (points) that a function can return.
• Domain: A set of all points over which a function is defined.
• Function: A mathematical formula that produces one and only one result for each input.

### Review of Domain, Range, and Functions

An input domain is defined in a previous section as the set of input values (x) for which a function is defined.  Defining a function includes defining its domain.  As an example, the domain of the function f(x)=√x is x≥0.

For a given input, a function’s range can be thought of as the set of acceptable solutions, or “output” values (y).  A function has only one result per domain by definition.  For instance, the function f(x)=x² has a range of f(x)≥0, because the square of a number always yields a positive result.

In terms of both domain and range, a function is any mathematical formula that produces one and only one output for each input. Therefore, every single domain value also has a single range value as a result, but not necessarily the other way around. The same x-value can have two different y-values, but each y-value must be accompanied by a distinct x-value.  Result values (y-values) can repeat, but input values cannot (x-values).

### Determining Domain and Range

Visualizing the domain and range can be done using a graph, such as the following red U-shaped curve for f(x) = x2. The blue N-shaped (inverted) curve is the graph of f(x)= −(1/12)x³.

Example 1: Determine the domain and range of each graph pictured below:

As both graphs continue on to the left (negative values) and to the right (positive values) for x (input values), all real numbers are included as inputs. Both graphs continue to infinity in both directions; therefore, the domain for both graphs is the set of all real numbers, denoted by R.

If we now look at the possible outputs or y-values, f(x), (looking up and down the y-axis, notice that the red graph does NOT include y-values that are negative, whereas the blue graph does include both positive and negative values. Therefore, the range for the graph f(x) = x², is R except for y<0, or simply stated: y≥0. The range for the graph f(x)=−(1/12)x³, is R.

How do I find the domain and range of a function?

How to Find The Domain and Range of an Equation? To find the domain and range, we simply solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain. To calculate the range of the function, we simply express x as x=g(y) and then find the domain of g(y).

How do I find the domain of a function?

• The input values should be identified.
• Since there is an even root, the negative roots should be excluded from the radicand. Set the radicand to greater than or equal to zero and solve for x.
• Solution(s) are the domain of the function. Consider writing your answer in intervals if possible.

What is the easiest way to find the domain and range?

Another way to identify the range and domain of functions is by using graphs. A graph’s domain consists of all the input values shown on the x-axis since the domain refers to all the possible input values. An output range is a set of possible values, which are shown on the y-axis.

What does it mean to find the domain?

According to this definition, the domain consists of all possible x-values that will cause the function to produce a real y value. In finding the domain, bear in mind that the denominator (bottom) cannot be zero. In this section, a positive number must appear under the square root symbol.