Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range coorespond with different values in comparison to one another. For instance, 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 a cumulative score of 81 - 90, and so on. Here, the grade changes with the result. In mathematical terms, 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 instance, a function can be specified as a machine that catches specific items (the domain) as input and makes certain other objects (the range) as output. This could be a instrument whereby you could get multiple treats for a respective amount of money.
Today, we review the basics of the domain and the range of mathematical functions.
What are 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 check 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. To put it simply, 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 cloud apply any value for x and acquire itsl output value. This input set of values is required to find the range of the function f(x).
Nevertheless, there are certain conditions under which a function must not be defined. For instance, if a function is not continuous at a certain point, then it is not specified 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 batch of all y-coordinates or dependent variables. So, working with the same function y = 2x + 1, we can see that the range will be all real numbers greater than or equivalent tp 1. No matter what value we plug in for x, the output y will always be greater than or equal to 1.
Nevertheless, just like with the domain, there are particular terms under which the range must not be defined. For instance, if a function is not continuous at a specific point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range could also be represented using interval notation. Interval notation explains a group of numbers applying two numbers that classify the bottom and higher bounds. For instance, the set of all real numbers in the middle of 0 and 1 might be identified applying interval notation as follows:
(0,1)
This reveals that all real numbers higher than 0 and less than 1 are included in this set.
Similarly, the domain and range of a function could be classified via interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) might be classified as follows:
(-∞,∞)
This reveals that the function is defined for all real numbers.
The range of this function can be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be represented with graphs. For example, let's consider the graph of the function y = 2x + 1. Before plotting a graph, we need to discover 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 look from the graph, the function is specified for all real numbers. This shows 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 process of finding domain and range values is different for different types of functions. Let's watch some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is stated for real numbers. For that reason, 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 as 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 returns 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 shifts between -1 and 1. In addition, the function is defined 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]
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 contains 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 includes 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 Questions on Domain and Range
Find 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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