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

What are Domain and Range?

In simple terms, domain and range apply to different values in in contrast to each other. For example, let's check out grade point averages of a school where a student gets an A grade for an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade shifts with the total score. Expressed mathematically, 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 instance, a function might be specified as a machine that catches particular pieces (the domain) as input and makes particular other items (the range) as output. This could be a machine whereby you can buy multiple items for a particular quantity of money.

In this piece, we review 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 cooresponds to the x-values and y-values. For instance, let's look at 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. To put it simply, it is the set of all x-coordinates or independent variables. So, let's consider the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we can apply any value for x and acquire a respective output value. This input set of values is required to discover the range of the function f(x).

But, there are particular terms under which a function may not be defined. So, if a function is not continuous at a certain point, then it is not defined for that point.

The Range of a Function

The range of a function is the set of all possible output values for the function. In other words, it is the group of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we can see that the range will be all real numbers greater than or equivalent tp 1. Regardless of the value we apply to x, the output y will continue to be greater than or equal to 1.

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

Domain and Range in Intervals

Domain and range might also be identified via interval notation. Interval notation expresses a batch of numbers using two numbers that classify the bottom and upper bounds. For example, the set of all real numbers among 0 and 1 could be identified working with interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function might be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be represented via graphs. For instance, let's consider the graph of the function y = 2x + 1. Before plotting a graph, we need to determine 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 look from the graph, the function is stated for all real numbers. This shows 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 task of finding domain and range values differs for various types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is stated for real numbers. Therefore, 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 can be a possible input value. As the function only returns 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 oscillates between -1 and 1. Also, the function is defined 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 structure y= √(ax+b) is defined only for x ≥ -b/a. For that reason, the domain of the function contains 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 includes all non-negative real numbers.

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

• Domain: [-b/a,∞)

• Range: [0,∞)

Practice Questions 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 would be happy to set you up with a private math instructor if you need assistance comprehending domain and range or the trigonometric subjects. Our Riverside math tutors are practiced educators who focus on work with you on your schedule and customize their instruction techniques to match your learning style. Contact us today at (951) 337-4564 to learn more about how Grade Potential can support you with reaching your academic goals.