Domain and Range - Examples | Domain and Range of a Function

What are Domain and Range?

In simple terms, domain and range coorespond with different values in in contrast to one another. For example, let's check out grade point averages of a school where a student receives an A grade for a cumulative score of 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade changes with the result. In mathematical terms, the result is the domain or the input, and the grade is the range or the output.

Domain and range could also be thought of as input and output values. For example, a function could be specified as a machine that catches respective objects (the domain) as input and produces certain other pieces (the range) as output. This can be a machine whereby you might buy several treats for a respective quantity of money.

Here, we discuss the basics of the domain and the range of mathematical functions.

What are the Domain and Range of a Function?

In algebra, the domain and the range cooresponds to the x-values and y-values. For instance, let's view the coordinates for the function f(x) = 2x: (1, 2), (2, 4), (3, 6), (4, 8).

Here the domain values are all the x coordinates, i.e., 1, 2, 3, and 4, whereas the range values are all the y coordinates, i.e., 2, 4, 6, and 8.

The Domain of a Function

The domain of a function is a set of all input values for the function. To clarify, it is the set of all x-coordinates or independent variables. For example, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we can apply any value for x and get a corresponding output value. This input set of values is required to find the range of the function f(x).

However, there are certain terms under which a function must not be stated. For example, if a function is not continuous at a particular point, then it is not specified for that point.

The Range of a Function

The range of a function is the group of all possible output values for the function. To be specific, it is the group of all y-coordinates or dependent variables. For example, working with the same function y = 2x + 1, we might see that the range is all real numbers greater than or equal to 1. Regardless of the value we apply to x, the output y will continue to be greater than or equal to 1.

Nevertheless, just as with the domain, there are specific terms under which the range may not be specified. For instance, if a function is not continuous at a particular point, then it is not stated for that point.

Domain and Range in Intervals

Domain and range could also be classified via interval notation. Interval notation indicates a set of numbers applying two numbers that identify the bottom and upper boundaries. For instance, the set of all real numbers among 0 and 1 might be classified working with interval notation as follows:

(0,1)

This means that all real numbers more than 0 and lower than 1 are included in this group.

Similarly, the domain and range of a function can be classified using interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) might be identified as follows:

(-∞,∞)

This means that the function is defined for all real numbers.

The range of this function could be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be identified with graphs. So, let's consider the graph of the function y = 2x + 1. Before charting a graph, we must discover all the domain values for the x-axis and range values for the y-axis.

Here are the coordinates: (0, 1), (1, 3), (2, 5), (3, 7). Once we graph these points on a coordinate plane, it will look like this:

As we could watch from the graph, the function is specified for all real numbers. This means that the domain of the function is (-∞,∞).

The range of the function is also (1,∞).

This is because the function generates all real numbers greater than or equal to 1.

How do you find the Domain and Range?

The process of finding domain and range values differs for different types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is specified for real numbers. For that reason, the domain for an absolute value function contains all real numbers. As the absolute value of a number is non-negative, the range of an absolute value function is y ∈ R | y ≥ 0.

The domain and range for an absolute value function are following:

• Domain: R

• Range: [0, ∞)

For Exponential Functions

An exponential function is written as y = ax, where a is greater than 0 and not equal to 1. For that reason, each real number could be a possible input value. As the function only delivers positive values, the output of the function contains all positive real numbers.

The domain and range of exponential functions are following:

• Domain = R

• Range = (0, ∞)

For Trigonometric Functions

For sine and cosine functions, the value of the function varies between -1 and 1. In addition, the function is specified for all real numbers.

The domain and range for sine and cosine trigonometric functions are:

• Domain: R.

• Range: [-1, 1]

Take a look at the table below for the domain and range values for all trigonometric functions:

For Square Root Functions

A square root function in the form y= √(ax+b) is stated only for x ≥ -b/a. Therefore, the domain of the function contains all real numbers greater than or equal to b/a. A square function will consistently result in a non-negative value. So, the range of the function contains all non-negative real numbers.

The domain and range of square root functions are as follows:

• Domain: [-b/a,∞)

• Range: [0,∞)

Practice Examples on Domain and Range

Discover the domain and range for the following functions:

1. y = -4x + 3

2. y = √(x+4)

3. y = |5x|

4. y= 2- √(-3x+2)

5. y = 48

Let Grade Potential Help You Master Functions

Grade Potential can connect you with a 1:1 math instructor if you need support understanding domain and range or the trigonometric subjects. Our San Gabriel math tutors are practiced educators who strive to tutor you on your schedule and tailor their teaching techniques to fit your learning style. Call us today at (626) 888-8501 to hear more about how Grade Potential can support you with obtaining your educational objectives.