What is a Function in Maths?

Function

A technical definition of a what is a function in math is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. Typical examples of function in math are from integers to integers or from the real numbers to real numbers.

In addition, it is a relation or a process which connects each element x of a set X to the domain of the function and to a single element y of another set Y (usually the same set), the codomain of the function.

Suppose if we call the function ‘f”, then we can denote this relation as y = f (x) (read f of x), the element x is the argument or input of the function, and y denotes the value of the function, the output, or the image of x by f. Let us study what is a function in math in detail.

Definition of Function

As told earlier it is an operation that links each element of a set X to a single element of a set Y. Formally, we can define a function f from a set X to a set Y by a set G of order pairs (x, y). Such that x \(\sum\)X, y  \(\sum\) Y, and all elements of X is the first component of exactly one order pair in G. The set G is the graph of the function. We generally distinguish a function from its graph.

The term X and Y in the definition of a function, are respectively the domain and the codomain of the function f. The two functions f and g will be equal if their domain and codomain sets are the same. Also, their output values agree on the whole domain. The range of a function represents the set of images of all elements in the domain. But, we sometimes use range as a synonym of the codomain.

Notation of Function

There are various standard ways for denoting functions. Below we will discuss them.

1) Functional notation: Functions are denoted by a symbol consisting generally of a single letter in italic font, most often the lower-case letters f, g, h. In the following notation (“y equals f of x”) y=f(x) means that the pair (x, y) belongs to the set of pairs defining the function f. If X is the domain of f, the set of pairs defining the function is thus, using set-builder notation, (x,f(x)): x  \(\sum\) X.

2) Arrow notation: For expressing domain X and the codomain Y of a function f, we often use the arrow notation (as “the function f from X to Y” or ” most importantly, the elements of X to elements of Y is mapped by a function f);

\(f:X \rightarrow Y\)

\(X \overset{f}{\rightarrow} Y\)

3) Index notation: It is often used instead of functional notation. So instead of writing f (x), one can write fx. This is commonly the case for those functions whose domain is the set of the natural numbers.

4) Dot notation: In the notation x  maps to f(x)  the symbol x does not represent any value, it is simply a placeholder meaning that, if the value of x is changed by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Hence, we can replace x by any symbol, often an interpunct ” ⋅ “. Example, we will write x maps to \(ax^{2}\)  as \( a(\cdot )^{2}\).

Representing a Function

1) A graph commonly gives an intuitive picture of a function. It helps to understand a function. Also, we can easily see from its graph, whether a function is increasing or decreasing.

2) We can represent a function as a table of values. If we consider the domain of a function is finite, then we can completely specify a function in this way. Also, if a domain of a function is continuous, a table can give the values of the function at specific values of the domain.

3) Often we use bar charts to represent functions whose domain is a finite set, the natural numbers, or the integers.

Solved Questions for You

Ques-1. What are the different types of function?

Answer: Basically, there are eight types of functions namely: linear, power, quadratic, polynomial, rational, exponential, logarithmic, and sinusoidal.

Ques-2. How do you identify a function?

Answer: It is relatively easy to determine whether an equation is a function by solving for y. When an equation is present along with a specific value for x, there should only be one corresponding y-value for that x-value. An example, x = y + 1 is a function because x will always be one greater than y.