What is a Function: Types, Graph & Composition

A function, in mathematics, denotes a special relationship between an element of a non-empty set with an element of another non-empty set. Meaning, from a set X to a set Y, a function is an assignment of an element of Y to each element of X, where set X is the domain of the function and the set Y is the codomain of the function.

• In mathematics, a function is simply a rule that allocates to each element of a set (called the domain) exactly one element of another set (called the range).
• These functions are usually represented by letters such as f, g, and h, etc, and the value of a function f at an element x of its domain is expressed by f(x).
• Moreover, a function is uniquely represented by the set of all pairs (x, f(x)), which is called the graph of the function.
• Functions are a fundamental part of Calculus in mathematics. 

There are several types of functions as well. The types of functions in maths are:

• Injective function or One to one function
• Surjective functions or Onto function
• Polynomial function
• Inverse Functions

Read More: Eccentricity

Key Terms: Domain, Calculus, Particular equation, Function, Into funtion, Onto Function, Polynomial function, Quadratic Functions, Constant Function

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What are Functions?

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A function is simply a relationship between an element of one non-empty set with one element of another non-empty set. If we expand the concept and try to understand it in simple terms, an equation is a function if, for any X in the domain of the equation (here domain is all possible values of X that can be plugged into the particular equation), the equation will produce exactly one value of Y when we evaluate the equation at a specific X.

Functions

For example, let us take two equations and try to find a function in them.

1. Y = X² + 3
2. Y²= X + 3

In the case of the first equation, for a given X, there is only one way to square it and 3 have to be added to get the result. So, no matter what value of X we put in the equation, there will be one possible value Y as we evaluate the equation at the value of X. Thus, the equation Y= X²+ 3, is a function.

In the case of the second equation, for the value X=6, the equation Y² = X+3 yields Y²=9. It means Y will be either 3 or -3. Thus, for one unique value of X=6, Y gives two values as per the second equation. Thus, this equation is not a function.

Relations and Functions Detailed Video Explanation

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What is a Function in Maths? 

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A function is a relation or the association of each element 'a' of set A which is a non-empty set, to at least to a single element 'b' of another non-empty set B.

Thus, a relation of set A (the domain of the function) to another set B (the co-domain of the function) is formed that can be called a function.

It can be mathematically transcribed as:

• A relation is described as a function if every element of set A has one and only one image in set B.
• A function is a relation from a non-empty set B to the domain of a function is A and no two distinct ordered pairs in f can have the same first element.
• A function from A → B and (a,b) ∈ f, then f(a) = b, where 'b' is the image of 'a' and 'a' is the pre-image of 'b'. Here the set A is called the domain of the function and set B is to be called the co-domain of it.

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Composition of Functions

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Assume that f : A → B and g: B→ C are two functions.

Fo this, the composition of f and g can be expressed as f(g) and it can be shown as the function f ∘∘ g = f(g(x)) for x ∈ A.

The two functions are f(x) and g(x).

Consider that f(x) = (x+1) and g(x) = x². In case of that,

f ∘∘ g = f(x²)

→ f(x²) = x² + 1

g ∘∘ f = g(f(x))

→ g(f(x)) = g((x+1))

= (x+1)² = x²+ 2x + 1

Hence, we can say that the composition of functions is not commutative. f ∘∘ g ≠ g ∘∘ f

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Domain, Co-domain and Range

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There are two major concepts needed to be considered while we talk about functions are ‘domain’ and ‘range’.

• A domain is the set of values that can be plugged in as input in the function while it can be defined.
• A Co-domain is the set of values that have the potential to come out as output of a function.
• The range is all the values that can be obtained as the output of the function involved.

Identification of a Function in Math

A function in math can simply be signified as the correspondence from one value x of the first set to another value y belonging to the second set. This correspondence can be of any of the four types:

• 1 to 1
• Many to 1
• 1 to Many
• Many to Many

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Examples of Function

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From the above-written explanation, we can interpret that whenever a variable quantity Y is a function of another variable quantity X, it means Y depends on X; thus, the value of Y can be determined by plugging any value of X.

This dependence is written as, Y=f (X)

• So, the area(A) of a circle can be expressed in terms of its radius(r), A = π r². Here it can be said that area A is dependent on the radius r. In the terms of functions, we say that A is a function of r.
•  Let’s assume volume V of a sphere is a function of its radius(r). Then the dependence of V on r can be written as V = 4/3 π r³.
• The acceleration of a body, a, with fixed mass m can be indicated as a function of the force F applied on the body. Thus, a = F/m.

Also Read: Midpoint Formula

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Type of Functions

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The types of functions can be defined on the basis of the domain, range, and function expression. There are numerous types of functions in mathematics. Some of them are mentioned below:

One to One function

Also known as an injective function, a one-to-one function is defined by f: A→ B such that every element of set A is connected to a distinct element in set B. Here, for the given function- every element of one domain has a distinct image or co-domain element.

Many to one function

It can be defined by the function f: A → B, such that more than one element of set A is connected to the same element in set B. In this case, more than one element has the same co-domain or image.

Onto functions

In the case of an onto function, every element in the codomain is related to the domain element. For a function defined by f: A → B, such that every element in set B must have a pre-image in set A.

Into Function

Here, there will be certain elements in the co-domain that would not have any pre-image. The elements in set B are in excess and not connected to any of the elements in set A. The into the function is exactly opposite in properties to an onto function.

Constant Functions

It is the kind of function which gives the same value of output for any input. It is expressed as, f(x) = c, c =constant.

For example, f(x) = 4 is a constant function. Whatever we put as x we get 4 as output.

Polynomial functions

It is a type of function which can be expressed as a polynomial. Thus, assuming that P(x) = a_n xⁿ + a_n-1 x^n-1+.……….…+a₂ x² + a₁ x + a₀, then for x ≫ 0 or x ≪ 0, P(x) ≈ a_n xⁿ. 

Thus, polynomial functions are known to approach power functions for very large variable values.

Quadratic function

Here is a type of function which has the highest power of 2 in the polynomial function. It is expressed as, f(x)=a(x-h)²+k.

For example, f(x) = x² + 2 is a quadratic function

Read More: Differential Equation

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Representation of Functions in Math

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The rule that mentions a function can come in a variety of forms depending on how it is defined. It can be expressed as piecewise-defined functions or in terms of other formulas.

In case we express f(x) = √x, for x ≥ 0, then, here, the inputs can be denoted as numbers and the 'taking square root' function takes all non-negative real numbers, yielding the output as f(x). Most often it is a formula, as in:

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Function in Algebra

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A function is an equation for which any value of x that can be put into the equation will produce exactly one and only one output such as y. It is mathematically translated as;

y = f(x)

(Where, x is an independent variable and y is a dependent variable.)

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What is a Function on a Graph?

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It is done by placing the values of the independent variable on the X-axis and values of the dependent variable along the Y-axis in a cartesian plane. All the points of the graph must satisfy the function y = f(x). For instance:

Let us assume, y = x + 3

• when x = 0, y = 3
• when x = -2, y = -2 + 3 = 1
• when x = -1, y = -1 + 3 = 2
• when x = 1, y = 1 + 3 = 4
• when x = 2, y = 2 + 3 = 5

Algebra of Functions

Algebra of functions refers to the rules and operations that can be applied to functions in algebraic expressions. In case of functions f(x) and g(x), where f: X → R and g: X → R, where x ϵ X, we have:

• (f-g) (x) = f(x) - g(x)
• (f+g) (x) = f(x) + g(x)
• (f.g) (x) = f(x) .g(x)
• (k f(x)) = k (f(x)), here, k = real number
• (f/g)(x) = f(x) /g(x), here, g(x) ≠ 0

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Things to Remember

• A function is referred to as a process or relationship between an element of one non-empty set with one element of another non-empty set.
• In other words, a function is a relation or the association of each element 'a' of set A which is a non-empty set, to at least to a single element 'b' of another non-empty set B.
• A relation of set A (the domain of the function) to another set B (the co-domain of the function) is formed which can be called a function.
• A domain refers to the set of values that can be plugged in as input in the function while it can be defined.
• A Co-domain is a set of values that have the potential to come out the output of a function.
• The range is all the values that can be obtained as the output of the function involved.
• The area (A) of a circle can be expressed in terms of its radius(r), A = πr²

Read More:

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Previous Year Questions 

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