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 comparison 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 a cumulative score of 81 - 90, and so on. Here, the grade shifts with the average grade. In mathematical terms, 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 can be defined as an instrument that takes particular pieces (the domain) as input and makes particular other items (the range) as output. This might be a instrument whereby you can obtain different snacks for a specified quantity of money.
Today, we discuss 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 indicate the x-values and y-values. For example, 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, 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 put it simply, 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) can 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 find the range of the function f(x).
However, there are specific cases under which a function cannot be stated. For example, if a function is not continuous at a certain point, then it is not stated for that point.
The Range of a Function
The range of a function is the batch of all possible output values for the function. To be specific, it is the group 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 equivalent tp 1. No matter what value we apply to x, the output y will always be greater than or equal to 1.
However, just like with the domain, there are specific terms under which the range must not be specified. For instance, if a function is not continuous at a specific point, then it is not defined for that point.
Domain and Range in Intervals
Domain and range might also be classified using interval notation. Interval notation expresses a batch of numbers working with two numbers that represent the lower and upper limits. For example, the set of all real numbers among 0 and 1 could be classified using interval notation as follows:
(0,1)
This means that all real numbers greater than 0 and lower than 1 are included in this set.
Similarly, the domain and range of a function could be identified via interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be represented as follows:
(-∞,∞)
This means that the function is specified for all real numbers.
The range of this function might be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be identified using graphs. For instance, let's review the graph of the function y = 2x + 1. Before creating 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 plot these points on a coordinate plane, it will look like this:
As we might look from the graph, the function is stated for all real numbers. This means that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
This is because the function produces all real numbers greater than or equal to 1.
How do you find the Domain and Range?
The process of finding domain and range values differs for different types of functions. Let's take a look at some examples:
For Absolute Value Function
An absolute value function in the form y=|ax+b| is defined 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:
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Domain: R
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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 might be a possible input value. As the function just delivers positive values, the output of the function consists of all positive real numbers.
The domain and range of exponential functions are following:
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Domain = R
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Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function alternates among -1 and 1. Further, the function is stated for all real numbers.
The domain and range for sine and cosine trigonometric functions are:
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Domain: R.
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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 form y= √(ax+b) is defined only for x ≥ -b/a. For that reason, the domain of the function consists of 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 consists of all non-negative real numbers.
The domain and range of square root functions are as follows:
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Domain: [-b/a,∞)
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Range: [0,∞)
Practice Examples on Domain and Range
Discover the domain and range for the following functions:
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y = -4x + 3
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y = √(x+4)
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y = |5x|
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y= 2- √(-3x+2)
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y = 48
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