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 consider the grading system of a school where a student receives an A grade for a cumulative score of 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade adjusts with the result. Expressed mathematically, 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 example, a function might be defined as a tool that catches respective pieces (the domain) as input and produces specific other items (the range) as output. This could be a machine whereby you might get several treats for a respective quantity of money.
In this piece, we discuss 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. So, 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. In other words, it is the set of all x-coordinates or independent variables. For example, let's review the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we cloud plug in any value for x and acquire a corresponding output value. This input set of values is required to find the range of the function f(x).
Nevertheless, there are certain cases under which a function may not be stated. So, if a function is not continuous at a particular 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. To be specific, it is the set of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we could see that the range would be all real numbers greater than or equivalent tp 1. No matter what value we apply to x, the output y will continue to be greater than or equal to 1.
However, just as with the domain, there are certain terms under which the range must not be defined. For instance, if a function is not continuous at a particular point, then it is not defined for that point.
Domain and Range in Intervals
Domain and range might also be identified using interval notation. Interval notation indicates a set of numbers using two numbers that represent the bottom and higher bounds. For instance, the set of all real numbers among 0 and 1 could be classified using interval notation as follows:
(0,1)
This denotes that all real numbers more than 0 and less than 1 are included in this set.
Similarly, the domain and range of a function might be identified by applying interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) can be identified as follows:
(-∞,∞)
This reveals that the function is stated for all real numbers.
The range of this function might be classified as follows:
(1,∞)
Domain and Range Graphs
Domain and range could also be classified with graphs. So, let's consider the graph of the function y = 2x + 1. Before charting a graph, we must 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 can 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,∞).
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 take a look at some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is specified for real numbers. For that reason, the domain for an absolute value function consists of 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, each real number could be a possible input value. As the function only returns 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 shifts between -1 and 1. In addition, 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 form y= √(ax+b) is stated only for x ≥ -b/a. Consequently, the domain of the function includes 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:
-
y = -4x + 3
-
y = √(x+4)
-
y = |5x|
-
y= 2- √(-3x+2)
-
y = 48
Let Grade Potential Help You Excel With Functions
Grade Potential can connect you with a one on one math instructor if you are looking for support comprehending domain and range or the trigonometric concepts. Our Tampa math tutors are skilled professionals who focus on partner with you when it’s convenient for you and customize their tutoring techniques to suit your needs. Reach out to us today at (813) 686-6467 to hear more about how Grade Potential can assist you with achieving your learning goals.
