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# What is a Function?

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Last updated date: 13th Aug 2024
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## Functions

What do you understand by the term function? It is nothing but a duty or job one has to play. Like for instance, in day to day life, you can see individuals who play their job roles. Similarly, we have functions in mathematics as well. Let’s learn about them.

## What is the Function in Maths?

If you are wondering what is a function, it can be defined as a unique relationship between each element of one set with only one element of another set. However, both sets must be non-empty. F(x) is the traditional way to express function.

You can also say it as a machine having both input and output and some of the other ways they are related.

If we consider the two sets to be A, and B, respectively, we can write:

f:  A -> B is considered a function when a ∈ A, a unique element is also there for b ∈ B, provided (a, b) ∈ f.

Here, we have provided a simple example of an input-output relationship, making it easy for you to understand what is a function.

## Example: Multiplication with 3

 Input Output Relationship 0 0 x 3 1 3 x 3 2 6 x 3 .... .... x 3

Some other function examples:

• Functions like cosine, sine and tangent have a usage in trigonometry.

• x2.

• x3 + 1, etc.

## How to Name a Function?

Now that you are familiar with what is a function, you must know how to name it. As said earlier that conventionally ‘f’ is used, but the alphabet ‘g’ can also work. Sticking on to the conventional name, we can write the following example of a function expression:

f(x) = x2

Here, f = function name.

x = input.

x2 = output.

So, if we put a value of x (suppose 4), the output obtained is 16. It can be expressed as f(4) = 16.

## Kinds of Functions

### One to One Function

When elements of set A have a separate component of set B, we can determine that it is a one to one function. Besides, you can also call it injective.

### Many to One Function

As the name suggests, here more than two elements in set A are mapped with one element in set B.

Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.

Also, if a function is both one to one and onto function, it is known as bijective. This means, all the elements of A are mapped with separate elements in B, and A holds pre-image of elements of B.

### Widely used Functions

The formulas that we use while solving various mathematical problems are expressions of known functions. Like for instance, if we take the area of a circle formula a = pi x r2, it gives independent variable r and dependent variable A. This can also be termed as a linear function containing one dependent and one independent variable. A linear function can be represented in the form of y = f(x) = a + bx

Where a = constant term or y-intercept.

b = slope or coefficient of the independent variable.

Furthermore, exponential functions have extensive usage in mathematics but not more than linear functions. In order to find out an exponential function, the independent variable is allowed to be the exponent.

The above image shows the increase of exponential function. They act as solutions for simple dynamical systems, for example, the growth of bacteria.

### Do It Yourself

1. f(x1) = f(x2) => x1 = x2 ∀ x1 x2 belongs to A. What is function is A->B?

(a) one - one (b) onto (c) one - one and onto (d) many – one

By now we presume that you have well understood what you mean by function, what is a linear function and what is exponential growth. If you wish to gather more knowledge about other functions like what is an objective function, rational function, modulus function, etc. we recommend you to download the Vedantu app.

## FAQs on What is a Function?

1. What are the Kinds of Functions in Mathematics?

Ans. Various types of functions are there in Maths like one to one (injective), many to one, onto (surjective), into, polynomial, linear, identical, quadratic, rational, cubic, modulus, signum, greatest integer, etc.

2. How to Determine Whether a Relation is a Function or Not?

Ans. You must remember that every function is considered a relation, but vice versa may not be possible always. For instance, consider a set of ordered pairs {(1, 5), (8, 4), (3, -2), (1, 0), (6, -3)}. Each of the input values must hold one output value, and if the same is found out, then this provided relation is a function. Students are supposed to sketch mapping diagrams and figure out the domain and range.

3. Why is a Circle, Not a Function?

Ans. A function can be defined as the mapping between two sets, while a circle is a group of points on a plane. Both things are entirely different.

When you map the coordinates of ordered pairs, each value in the domain is related to one point in the codomain. However, when a line passes through a circle, it cuts the circle at two points.