Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In simple terms, domain and range apply to multiple values in in contrast to one another. For example, let's consider grade point averages of a school where a student earns 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 adjusts with the average grade. In math, the total is the domain or the input, and the grade is the range or the output.
Domain and range can also be thought of as input and output values. For instance, a function could be stated as an instrument that catches specific objects (the domain) as input and produces certain other objects (the range) as output. This can be a instrument whereby you could get different treats for a respective quantity of money.
Today, we review the essentials 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, 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 group of all input values for the function. In other words, 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) might be any real number because we can apply any value for x and obtain a respective output value. This input set of values is needed to figure out the range of the function f(x).
Nevertheless, there are certain terms under which a function must not be specified. For example, if a function is not continuous at a specific point, then it is not stated 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 set of all y-coordinates or dependent variables. For example, using the same function y = 2x + 1, we can see that the range will be all real numbers greater than or equal to 1. Regardless of the value we apply to x, the output y will always be greater than or equal to 1.
But, as well as with the domain, there are particular terms under which the range must not be defined. For example, if a function is not continuous at a specific point, then it is not specified for that point.
Domain and Range in Intervals
Domain and range can also be represented using interval notation. Interval notation explains a batch of numbers using two numbers that identify the lower and higher bounds. For example, the set of all real numbers in the middle of 0 and 1 can be represented working with interval notation as follows:
(0,1)
This means that all real numbers greater than 0 and lower than 1 are included in this set.
Equally, the domain and range of a function could be identified by applying interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) might be represented as follows:
(-∞,∞)
This means that the function is defined for all real numbers.
The range of this function can be classified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be identified via graphs. For instance, let's consider the graph of the function y = 2x + 1. Before plotting a graph, we must 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 chart these points on a coordinate plane, it will look like this:
As we can look 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 find 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 specified for real numbers. Therefore, 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 in the form of y = ax, where a is greater than 0 and not equal to 1. For that reason, every real number might be a possible input value. As the function only produces 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. Further, the function is stated 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 structure y= √(ax+b) is specified just 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 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 Examples on Domain and Range
Discover the domain and range for the following functions:
-
y = -4x + 3
-
y = √(x+4)
-
y = |5x|
-
y= 2- √(-3x+2)
-
y = 48
Let Grade Potential Help You Master Functions
Grade Potential can pair you with a one on one math tutor if you need support understanding domain and range or the trigonometric subjects. Our San Antonio math tutors are practiced educators who focus on work with you when it’s convenient for you and customize their tutoring methods to fit your needs. Contact us today at (210) 879-9877 to hear more about how Grade Potential can help you with achieving your educational objectives.
