Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range refer to different values in in contrast to each other. For instance, let's consider the grade point calculation of a school where a student earns an A grade for a cumulative score of 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade shifts with the result. Expressed mathematically, the score 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 can be specified as an instrument that catches specific items (the domain) as input and generates particular other pieces (the range) as output. This might be a tool whereby you could get multiple items for a particular quantity 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. For instance, 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, for 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. To clarify, it is the set of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we cloud apply any value for x and obtain itsl output value. This input set of values is needed to figure out the range of the function f(x).
Nevertheless, there are certain conditions under which a function cannot be stated. For instance, 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 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, applying the same function y = 2x + 1, we could see that the range will be all real numbers greater than or equal to 1. No matter what value we plug in for 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 may not be stated. For example, if a function is not continuous at a particular point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range can also be identified via interval notation. Interval notation explains a group of numbers working with two numbers that identify the lower and higher boundaries. For example, the set of all real numbers in the middle of 0 and 1 can be identified applying interval notation as follows:
(0,1)
This reveals that all real numbers more than 0 and lower than 1 are included in this batch.
Similarly, the domain and range of a function can be represented with interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) could be classified as follows:
(-∞,∞)
This tells us that the function is defined for all real numbers.
The range of this function might be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range can also be identified using graphs. For instance, let's review the graph of the function y = 2x + 1. Before charting a graph, we have 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 plot these points on a coordinate plane, it will look like this:
As we could look 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 determine the Domain and Range?
The task of finding domain and range values is different for multiple 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. Consequently, 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 in the form of y = ax, where a is greater than 0 and not equal to 1. For that reason, any real number could 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 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 form y= √(ax+b) is defined only for x ≥ -b/a. Therefore, 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 Questions on Domain and Range
Find 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 match you with a private math instructor if you are looking for help mastering domain and range or the trigonometric topics. Our Denver math tutors are experienced professionals who aim to tutor you on your schedule and tailor their teaching strategy to match your needs. Call us today at (303) 945-3709 to learn more about how Grade Potential can assist you with obtaining your learning goals.
