Functions and their Representations

Introduction 

A function is an idealisation of varying quantities that depend on other quantities. A function is a mathematical domain or element that is defined as the form of set X to another set Y. This represents an assignment of an element of Y to each element of X. A function is a special type of relationship as we can say that a relation f from a set X to another set Y is called a function as if every element of Set X has one and only one image of the other set Y and no distinct elements of the other set Y. Further, they would have the same mapped first element. Also, X and Y are non-empty sets. The set X is a domain, and the whole set Y is a codomain.

Representations and Functions

A function is a particular type of relationship. A function is represented as f:X->Y, which is also written as f(x) = y, where (x,y) belongs to f and x belongs to X and y belongs to Y, which means the elements belong to their corresponding sets. For any function, the notation is always f(x), but at the place of x, there can be other variables depending on the circumstances of the set and question.

Representation of Functions

A function is represented in four forms, which are:

• Numerical representation: In this representation, the most basic way is the tabular form of representation, wherein the function and its properties are represented in a tabular manner. The table contains two columns and rows depending on the number of variables and the behaviour of the function so that we can say that this is variable independent. Through the table, we can extract the values, called the outputs. They should be infinite numbers to find the value that we want or are searching for. However, this is a difficult way of representation and solving problems using this for a large number of values can be difficult, especially to find the value of a particular variable.
• Graphical representation: This representation is also called visual representation. The way of representation of this type of function is easy to draw and understand as the input values are marked at the x-axis, and output values are present at the y axis. Further, for any input value, the output value is vertically displaced at the x-axis. For example, we can say that at x=a, the output will be equal to f(a). Here the graph shows the following properties of the function:

1. Where the graph increases or decreases so that we will be able to know about the function by this action
2. The point at which the rate of change is less or more in the function
3. The points of extreme values in the function.

Thus the graphs are easy and beneficial to know about the function’s behaviour and all the drawbacks that it contains.

• Verbal representation: This representation is shown by using words, for example, we say that for the input x, the function will give the largest integral value or smaller or equal to the value of x, which depend on its nature. This is called floor function, and when for the input of x, the function gives equal value to x, then this is called an identify function.
• Algebraic representation: This representation refers to the expression of the function using a mathematical model or the equation as we get to know the process of interpreting real-world problems and how to manage them conveniently and easily. All the inputs and outputs are represented with proper measurements and reference of the variables, which includes accurate units reference diagram and other parameters. The input and output diagrams often use mathematical models. This representation is one of the most common representations. The functions are represented using formulae and denoted by lower case alphabets. For example: f: x -> x^4,

where x is a formula function, and f is the name of the function.

So with consideration to the pros and cons of these representations, and based on our preference, we should choose the representation of the particular function.

Define Representations and Functions

A function is defined as a rule that is assigned with one output value to every input value. The function is denoted as f(x), which is read as “f of x”. Further, we can say that a function is a relation that makes the element of one set to another set or links the elements of set x to set y. A function is a rule that always allows and relates the input to only one output.

The function is represented as f:X->Y, which is also written as f(x) = y, where (x,y) belongs to f and x belongs to X and y belongs to Y. In the representation, we see that every element of the set has its image, which is unique and distinct. For example, if we lift our hand towards the top, then waving our hand is a function, and so is moving in a circular motion near a street. This is an example that is related to the real world as a function.

Conclusion 

From the above content, we get to know both the functions and their representations. In short, a function is a special type of relationship where a relation f from a set X to the other set Y is called a function as if every element of Set X has one and only one image of the other set Y and no distinct elements of the other set Y. Further, they will have the same mapped first element. Moreover, it is represented as f:X->Y, which is also written as f(x) = y, where (x,y) belongs to f and x belongs to X and y belongs to Y. Finally, there are different ways of representation of functions.