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What is a Function?

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A Function in maths is a relation between two sets of values in which one set is the input value and the other set is the output value and each input value gives a particular output value. We represent a function in maths as, y = f(x) where x is the input value and for each x we get an output value as y.

In this article, we will learn about, functions in mathematics, their various types, examples, and others in detail.

What is a Function in Maths?

A function in mathematics is a relation between the input values (domain) and the output values (range) of the given sets such that no two variables from the domain sets are linked to the same variable in the range set. A simple example of a function in maths is f(x) = 2x which is defined on R→R, here any variable in the domain is related to only one variable in the range.

A function in mathematics has a domain, codomain, and range. The domain is the set of all the possible values of x and the range of the function is the set of all the output values of y. We can also say that a function in maths is a relation with a unique output and no two input value has similar output in a function which is not the case for relation.

Function Definition

A relation or method connecting each member of set A to a unique member of set B  via a defined relation is called the Function. Set A is called the domain and set B is called the co-domain of the function. A function in mathematics from set A to set B is defined as,

f = {(a,b)| ∀ a ∈ A, b ∈ B}

Every function is a relation but every relation is not a function and the criteria for any relation to be considered a function as in function every element of set A has only one image in set B while in relation an element of set A can have more than one image in set B.

We define a functions in maths from non-empty set A to non-empty set B such that,

(a, b) ∈ f, then f(a) = b

where we called b as the image of a defined under the relation f.

Every element ‘a’ of set A has a unique image ‘b‘ in set B then it is a function.

Examples of Functions

A function in mathematics f is defined as, y = f(x) where x is the input value, and for each input value of x, we get a unique value of y. Various examples of the functions in maths defined on R→R are,

  • y = f(x) = 3x + 4
  • y = f(x) = sin x + 3
  • y = f(x) = -3x2 + 3, etc

Representation of Functions in Math

We represent a function in mathematics as,

y = f(x) = x + 3

Here, the set of values of x is the domain of the function and the set of output values of y is the co-domain of the function. Here, the function is defined for all real numbers as it gives a unique value for each x but it is not always possible to get the output for each value of x in such case we define the function in two parts, this can be understood as

  • f(x) = 1/(x – 2), where x ≠ 2
  • f(x) = x2 where x ∈ {R}

We can define a function in mathematics as a machine that that takes some input and gives a unique output. The function f(x) = x2 is defined below as,

Function in Maths

We can represent a function in math by the three method as,

  • Set of Ordered Pairs
  • Table Form
  • Graphical Form

For instance, if we represent a function as, “f(x) = x3 ” 

Another way to represent the same function is as the set of ordered pairs as,

f = {(1,1), (2,8), (3,27)}

In the above-mentioned set, the domain of the function is D = {1, 2, 3} and the range of the function is R = {1, 8, 27}

Domain and Range of Function f(x) = x^3

Types of Function

Different Types of Functions are used to solve various types of mathematical problems especially related to curves and equations. We have five functions in mathematics that are based on the element mapping from set A to set B.

Injective function or One to One Function

The function in which each element of the domain has a distinct image in the codomain is called the Injective or One-to-One function.

f: A → B is said to be one-to-one or injective if the images of distinct elements of A under f are distinct, i.e.,

f(a1) = b1, f(a2) = b2

where a1, a2 ∈ A and b1, b2 ∈ B

Surjective functions or Onto Function

The function in which every element of codomain has a pre-image in the domain is called the Surjective function or Onto Function i.e. each element of codomain is associated with each element of the domain. No element of codomain should have an empty relation. The number of elements of codomain and range is the same.

f: A → B is said to be onto, if every element of B is the image of some element of A under f, i.e., for every b ϵ B, there exists an element ‘a’ in A such that f(a) = b.

Bijective Function

If a function has properties of both Injective(One to One) and Surjective(Onto function) then the function is called a Bijective Function. In Bijective Function, each element of the domain is related to each element of the codomain and also there is one-to-one relation. This implies that number of elements of the codomain and range are the same and no element either in the domain or codomain has empty relation.

Polynomial Function

The function in which the exponents of algebraic variables are non-negative integers is called a Polynomial Function. If the power of the variable is 1 it is called a linear function, if the power is 2 it is called a quadratic function, and if the power is 3 it is called a cubic function. Some examples of polynomial functions are mentioned below:

  • y = x2
  • y = 2x + 3
  • y = 3x3

Inverse Function

The function containing the inverse of another function is called the inverse function. Let’s say we have a function y = f(x) then its inverse function will be x = f-1(y). In y = f(x), the domain is x and the range is y while in the case of x = f-1(y), the domain is y and the range is x. Thus we can say that the domain of the original function is the range of its inverse function and the range of the original function is the domain of the original function. Some examples of inverse functions are,

  • y = tan-1(x)
  • y = x-1

What is a Function in Algebra?

A function in algebra is an equation for which any x that can be put into the equation will produce exactly one output such as y out of the equation. It is represented as y = f(x), where x is an independent variable and y is a dependent variable.

For example:

  • y = 2x + 1
  • y = 3x – 2
  • y = 4y
  • y = 5/x

Composition of Functions

If f: A → B and g: B→ C be two functions. Then the composition of f and g is denoted as f(g) and it is defined as the function fog = f(g(x)) for x ∈ A.

Let’s take two functions f(x) = x + 3 and g(x) = 2x2

fog = f(g(x))

fog = f(2x2)

fog = 2x2 + 3

Learn More, Composition of Function

Algebra of Functions

There are various operations that can be performed on the function and the algebra of the function for two functions f(x) and g(x) defined on the real value of x are,

  • (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)); {For, k is a real number}
  • (f/g)(x) = f(x) /g(x); {For g(x) ≠ 0}

What is a Function on a Graph?

A function can be easily reprent on a graph and in general any function represent a curve (including straight line) in the x-y plane.

To plot a function on a first find some points that lies on the function and then join these points according to the locus of the function. For example to graph the function (straight line) f(x) = y = 5x – 2 we need some point on the graph. To find the point the point on the graph we first take the random values of x and then find their corresponding values of y, as,

f(x) = y = 5x- 2

if x = 0, y = 5(0) – 2 = -2 ⇒ (x, y) = (0, -2)

if x = 1, y = 5(1) – 2 = 3 ⇒ (x, y) = (1, 3)

if x = 2, y = 5(2) – 2 = 8 ⇒ (x, y) = (2, 8)

Now joining these point we can get the graph of the function y = 5x – 2

Graphing Functions

Knowing the values of x allows a function f(x) to be represented on a graph. Because y = f(x), we can find the associated value for y by starting with the values of x. As a result, we can plot a graph in a coordinate plane using x and y values. Consider the following scenario:

Assume y = x + 3

When x = 0, y = 3

Similarly,

  • x = -2, y = -2 + 3 = 1
  • x = -1, y = -1 + 3 = 2
  • x = 1, y = 1 + 3 = 4
  • x = 2, y = 2 + 3 = 5
  • x = 3, y = 3 + 3 = 6

As a result, we may plot the graph for function x + 3 using these values.

Graph of Function for y = x + 3

Common Functions

Some Common Function that we observed regularly are,

Linear function: General Form y = mx + c

Quadratic Function

General Form of quadratic function is, ax2 + bx + c = 0

Cubic Function

General Form ax3 + bx2 + cx + d = 0

Apart form this there are various other common function that are,

Area of Circle Function

Area of Circle (A) is a function of its radius(r) such that,

A = πr2

Area of Triangle Function

Area of Triangle (A) is a function of its base(b) and height(h) such that,

A = (bh)/2

Some Other Functions are,

Complex Functions

Any function in which the input variable are complex function are called the complex function. A complex number is a number that can be plot on the complex plane. In a complex number we have real number and imaginary number. A complex number(z) is represented as, z= x + iy and a complex function is represented as, f(z) = P(x, y) + iQ(x, y)

Applications of Functions

When we say that a variable quantity y is a function of a variable quantity x, we indicate that y is dependent on x and that y’s value is determined by x’s value. This dependency can be expressed as follows: f = y (x).

  • The radius of a circle can be used to calculate the area of a circle. The radius r affects area A. We declare that A is a function of r in the mathematic language of functions. We can write A = f(r) =π×r2
  • A sphere’s volume V is a function of its radius. V = f(r) = 4/3×r3 denotes the dependence of V on r.
  • Force is a function of the acceleration of a body of fixed mass m. F = g(a) = m×a.

Read More,

Examples on Function

Example 1: For two functions f and g are defined as, f(x) = x2 and g(x) = ln(2x). Find the composite function (gof )( x )

Solution:

Given:

  • f(x) = x2
  • g(x) = ln(2x)

(gof )( x ) = g (f (x))

[g (f (x)] = ln(2f(x))

= ln(2x2)

= 2 ln(√2x)

Thus, (gof)(x) = 2 ln(√2x)

Example 2: Find the output of the function g(t)= 6t2 + 5 at

  • (i) t = 0
  • (ii) t = 2

Solution:

Given Function,

g(t)= 6t2 + 5t

  • (i) t = 0

g(0) = 6(0)2+5(0) = 0 + 0

g(0) = 0

  • (ii) t = 2

 g(2) = 6(2)2+5(2)

g(2) = 24 + 10

g(2) = 34

Example 3: The length of a rectangle is five times its breadth, express the area of the rectangle as a function of its length.

Solution:

Let, length of the rectangle be l and the breadth of the rectangle is, b

Now,

  • b = l/5

Area of Rectangle(A) = l × l/5 = l2/5

Thus, area of rectangle as the function of its length is,

A(l) = l2/5

Functions in Maths – FAQs

1. What is the Definition of a Function?

A relation f defined on a set A to another set B is called a function in math if each value of A has a unique value in set B.

2. How to Write a Function in Math?

The function f in mathematics is represented as f: A → B and is defined as, f(x) = x + 2. Here, for each unique value of x, we have a unique value of y.

3. How to Transform a Function?

We can easily transfrom a function to other functions by simply performing basic algebric operations on the function. The different tranformation of the functiona are, reflection, translation, rotation, etc.

4. What is a Rational Function?

A fraction function where the numerator and the denominator are polynomial functions is called the rational function. Some examples of the rational function are, 

  • f(x) = x2/(2x + 3)
  • g(x) = (6x + 3)/(x – 1), etc.

5. What is a Linear Function?

An algebraic function in which each term of the function is either constant or has a power of one is called a linear function. Some examples of the linear function are,

  • f(x) = 2x + 3
  • g(x) = x – 5, etc.

6. What are Domain and Codomain of a Function?

If we define the function as, y = f(x). Then the domain of the x is all the values of x for which y results in a unique value. And the co-domain of y is the set of all the values of y for each value of x.

7. How do you Identify a Function in Maths?

If any input value(x) of the domain in a relation has more than one image (y) then these relation can never be a function. So if the value of x is repeated in the ordered pair then it is never a function.



Last Updated : 28 Dec, 2023
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