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

What are Domain and Range?

To put it simply, domain and range refer to several values in in contrast to one another. For instance, let's take a look at the grading system of a school where a student gets an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade changes with the result. Expressed mathematically, the result is the domain or the input, and the grade is the range or the output.

Domain and range might also be thought of as input and output values. For example, a function could be specified as a machine that takes particular items (the domain) as input and makes particular other pieces (the range) as output. This might be a machine whereby you might get different snacks for a respective amount of money.

Today, we will teach you the basics of the domain and the range of mathematical functions.

What is the Domain and Range of a Function?

In algebra, the domain and the range indicate the x-values and y-values. For example, 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, because 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. In other words, it is the batch of all x-coordinates or independent variables. For example, let's consider the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we can apply any value for x and acquire a corresponding output value. This input set of values is needed to discover the range of the function f(x).

But, there are certain cases under which a function cannot be defined. So, if a function is not continuous at a certain point, then it is not specified for that point.

The Range of a Function

The range of a function is the batch of all possible output values for the function. To be specific, it is the group of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we can see that the range would be all real numbers greater than or equal to 1. Regardless of the value we assign to x, the output y will always be greater than or equal to 1.

However, just as with the domain, there are certain terms under which the range cannot be stated. For example, if a function is not continuous at a particular point, then it is not specified for that point.

Domain and Range in Intervals

Domain and range might also be classified using interval notation. Interval notation expresses a group of numbers using two numbers that represent the lower and higher boundaries. For example, the set of all real numbers between 0 and 1 might be represented working with interval notation as follows:

(0,1)

This means that all real numbers higher than 0 and less than 1 are included in this batch.

Similarly, the domain and range of a function could 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 reveals that the function is stated for all real numbers.

The range of this function can be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be identified using graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting a graph, we need to find 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 plot these points on a coordinate plane, it will look like this:

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

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

That’s because the function creates all real numbers greater than or equal to 1.

How do you figure out the Domain and Range?

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

For Absolute Value Function

An absolute value function in the form y=|ax+b| is defined for real numbers. For that reason, the domain for an absolute value function includes 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. Consequently, every real number could be a possible input value. As the function only produces positive values, the output of the function consists of 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 oscillates between -1 and 1. Also, the function is specified for all real numbers.

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

• Domain: R.

• Range: [-1, 1]

Just see 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 specified just for x ≥ -b/a. Consequently, the domain of the function consists of all real numbers greater than or equal to b/a. A square function will always 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 Excel With Functions

Grade Potential would be happy to set you up with a 1:1 math teacher if you need support mastering domain and range or the trigonometric subjects. Our Sarasota math tutors are skilled educators who focus on work with you on your schedule and tailor their teaching strategy to fit your needs. Reach out to us today at (941) 202-6569 to hear more about how Grade Potential can help you with obtaining your learning goals.