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 each other. For example, let's check out grade point averages of a school where a student receives 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 total score. In math, 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 stated as an instrument that catches particular objects (the domain) as input and makes certain other objects (the range) as output. This can be a machine whereby you might buy multiple snacks for a specified amount of money.
Today, we will teach you the fundamentals 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 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 batch of all input values for the function. In other words, it is the group of all x-coordinates or independent variables. So, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we might plug in 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 particular terms under which a function cannot be specified. For example, if a function is not continuous at a specific point, then it is not defined 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, using the same function y = 2x + 1, we might see that the range will be all real numbers greater than or equivalent tp 1. No matter what value we assign 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 cannot be specified. 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 could also be identified using interval notation. Interval notation expresses a group of numbers applying two numbers that identify the lower and higher bounds. For instance, the set of all real numbers among 0 and 1 can be represented applying interval notation as follows:
(0,1)
This reveals that all real numbers more than 0 and lower than 1 are included in this set.
Equally, the domain and range of a function can be classified via interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) might be identified as follows:
(-∞,∞)
This means that the function is stated for all real numbers.
The range of this function could be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be identified using graphs. For instance, let's consider the graph of the function y = 2x + 1. Before charting 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 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,∞).
This is due to the fact that the function generates all real numbers greater than or equal to 1.
How do you determine the Domain and Range?
The task 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 defined 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. Consequently, every real number might be a possible input value. As the function just 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 among -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]
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. Therefore, the domain of the function consists of all real numbers greater than or equal to b/a. A square function 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:
-
y = -4x + 3
-
y = √(x+4)
-
y = |5x|
-
y= 2- √(-3x+2)
-
y = 48
Let Grade Potential Help You Excel With Functions
Grade Potential would be happy to match you with a private math teacher if you are interested in help mastering domain and range or the trigonometric concepts. Our Long Beach math tutors are practiced professionals who focus on work with you on your schedule and tailor their tutoring methods to fit your learning style. Call us today at (562) 553-8992 to hear more about how Grade Potential can help you with reaching your educational objectives.
