Range Of A Relation

A range of relations can be described as the set of numbers or alphabets that come out as solutions to processes applied on functions under some specific domain values. 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 the range of a function and how to find the range in functions or relations?

What Is The Range?

The range is the final set of values that can be a group of numbers or a group of alphabets that are obtained because of the dependency of a function on its domain. Every element of the domain values plays a very important role in the final range of the function because ultimately a range is the total outputs possible. 

To understand what is the range 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 The 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}.

Range of Exponential Functions

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

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.

Range = (0, ∞)

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 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]. Similarly, below are the range values of basic trigonometric functions.

Sinθ = [-1, +1]

Cosθ = [-1, +1]

Tanθ = (-∞, +∞)

Cotθ = (-∞, +∞)

Secθ = (-∞, -1] U [+1, +∞)

Cosecθ = (-∞, -1] U [+1, +∞)

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

The set of all the outputs derived out of a function is called its range. Consider a function f: A→ B, where f(x) = 3x and A and B each represent a set of natural integers. A holds the domain values, while B holds the co-domain values in this case. The range is then the output of this function. The range is defined as a group of numbers that are factors of 3.  We discussed many important sections regarding what is the range of a relation, and how to find the range of functions and relations.