Domain Of A Relation

A domain of a relation can be described as the set of numbers or alphabets that are eligible to get used in a function or relation without any modification. A domain and a range are two parts of a function or relation that work together and go hand in hand along with each other. The purpose of a domain is to represent input values. On the other hand, the goal of the range is to describe the outputs that are practically possible. This article gives a brief note on the domain of the relation. We will start by getting a basic understanding of what is a domain of a function and what is the difference between a function or relation?

What Is A Domain?

A domain is a common term that is used very often in the algebra of mathematics. To understand what is a domain at a much better level, let’s take one example. Suppose, there is a function and let’s assume that function to be an electronic machine such as a grinder or juicer. Now, what happens when you add peeled fruits to a grinder or juicer. The juicer grinds it and releases the peeled fruits in the form of juice that has different physical complexions if compared to the fruits. 

Like that only, a juicer or grinder is the function or relation upon which all the operations are going to be performed. The peeled fruits are the representation of domain values that we add in the function, whereas the juice is the representation of ranges of values that we receive after performing various operations on the functions. The range is the desired set of values obtained from multiple processes on the function when the domain values are inserted. 

How To Find Domain And Range?

Suppose X = {2, 4, 6, 8}, f: X → Y, where R = {(x,y) : y = x+2}. In this example, X is a set, and there are some numbers written inside curly brackets. These numbers separated by commas are the input values for the function f, where X tends to Y. 

Domain values are {2, 4, 6, 8}.

The range is the ultimate result if we put all our domain values in the function or relation provided. The relation speaks that y = x +2. So, the range values are 4, 6, 8, 10.

Range values are {4, 6, 8, 10}.

The domain and range of a function or relation are represented in roaster form as we saw in sets.

Domain and Range of Exponential Functions

An exponential function is denoted by y = ax where a should not be less than or equal to zero. 

And we know that, a is not the domain instead x is the domain. But, there is no restriction on the x, which means that the domain is not restricted to any specific set of regions or values. The domain of exponential functions is all real numbers from the number line. 

If we talk about the range of an exponential function y, y is dependent on the value of x and a. Since, a is a constant number like 2,5,8, etc we can say that the range y is dependent on the domain x. 

Doesn’t matter whatever domain (values of x) you choose, the range (values of y) can never be negative. Y > = 0.

Domain = Real numbers(R) and Range = (0, ∞)

Domain and Range of Trigonometric Functions

Some of the most commonly used trigonometric functions are sin and cosine. These functions namely sinθ and cosθ are unique because we know it very well that θ is any angle. And an angle i.e. θ can be anything any number any degree. Hence the domain of trigonometric function sinθ and cosθ is all real numbers.

Let’s think about the domain of sinθ and cosθ. If you remember the graph of these two trigonometric functions, the sin and cos wave spreads only along the x-axis, but it’s restricted on the y-axis between the coordinates [0.-1] on the negative Y and [0,1] on the positive Y.

Hence, the range of trigonometric functions sinθ and cosθ lies between [-1, 1].

Domain and Range Of Absolute Value Function

An absolute value function is the simplest function to understand the domain and range. 

Y = |ax + b| is the general form of any absolute function. We can see clearly that we can insert any number in the place of x. Hence, the domain of absolute value function is all real numbers.

The purpose of the function is to convert any negative entity such as negative numbers into a positive number. Hence, by this understanding we can easily conclude that the range of absolute value function is everything in the positive y-axis. 

Domain = Real numbers, integers, etc

Range = [0, ∞)

Conclusion

A function’s domain signifies “all the data” that go into a function or relation.  The domain is the single term given to represent the set of all the possible inputs. Consider this f(x) = 2x. You could easily think that the domain is merely the collection of natural numbers, and the results obtained are referred to as the range, when the values x = 1,2,3,4,… are supplied. We discussed many important sections regarding the domain of relation, what is a domain of a function and the difference between a relation and a function.