What does the word function stand for? By the word function, we understand the responsibility or role one has to play. What is the function of the leaves of plants – to prepare food for the plant and store them? What is the function of the roots of plants? They absorb water and other nutrients from the ground and supply it to the plants and help them stand erect. Can we say that everyone has different types of functions? Let’s learn about some types of function in mathematics!

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## Functions

We can define a function as a special relation which maps each element of set A with one and only one element of set B. Both the sets A and B must be non-empty. A function defines a particular output for a particular input. Hence, f: AÂ â†’ BÂ is a function such that for a âˆˆ A there is a unique element b âˆˆ B such that (a, b) âˆˆ**Â **f

**Browse more Topics under Relations And Functions**

- Relations
- Functions
- Types of Relations
- Representation of Functions
- Composition of Functions and Invertible Function
- Algebra of Real Functions
- Cartesian Product of Sets
- Binary Operations

## Types of Functions

We have already learned about some types of functions like Identity, Polynomial, Rational, Modulus, Signum, Greatest Integer functions. In this section, we will learn about other types of function.

### One to One Function

A function f:Â AÂ â†’ B is One to One if for each element of A there is a distinct element of B. It is also known as Injective. Consider if a_{1} âˆˆ A and a_{2} âˆˆ**Â **B, f is defined as f: A â†’ B such that f (a_{1}) = f (a_{2})

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**Many to One Function**

It is a function which maps two or more elements of A to the same element of set B. Two or more elements of A have the same image in B.

### Onto Function

If there exists a function for which every element of set B there is (are) pre-image(s) in set A, it is Onto Function. Onto is also referred as Surjective Function.

### One – One and Onto Function

A function, f is One – One and Onto or Bijective if the function f is both One to One and Onto function. In other words, the function fÂ associates each element of A with a distinct element of B and every element of B has a pre-image in A.

**Browse more topics under Relations and Functions**

**Relations and Functions**

- Binary Operations
- Cartesian Product of Sets
- Algebra of Real Functions
- Composition of Functions and Invertible Function
- Representation of Functions
- Types of Relations
- Functions
- Relations

## Other Types of Functions

A function is uniquely represented by its graph which is nothing but a set of all pairs of x and f(x) as coordinates. Let us get ready to know more about the types of functions and their graphs.

### Identity Function

Let **R **be the set of real numbers. If the function f: **R**â†’**R** is defined as f(x) = y = x, for x âˆˆ R, then the function is known as Identity function. The domain and the range being **R.** The graph is always a straight line and passes through the origin.

### Constant Function

If the functionÂ f: **R**â†’**R** is defined as f(x) = y =Â c, for x âˆˆ R and c is a constant in **R**, then such function is known as Constant function. The domain of the function f is **R** and its range is a constant, c**.** Plotting a graph, we find a straight line parallel to the x-axis.

### Polynomial Function

A polynomial function is defined by y =a_{0Â }+ a_{1}x + a_{2}x^{2} + … + a_{n}x^{n}, where n is a non-negative integer and a_{0}, a_{1}, a_{2},…,Â _{nÂ }âˆˆ **R**. The highest power in the expression is the degree of the polynomial function. Polynomial functions are further classified based on their degrees:

- Constant Function: If the degree is zero, the polynomial function is a constant function (explained above).
- Linear Function: The polynomial function with degree one. Such as y = x + 1 or y = x or y = 2x â€“ 5 etc. Taking into consideration, y = x â€“ 6. The domain and the range are
**R.**The graph is always a straight line.

Quadratic Function: If the degree of the polynomial function is two, then it is a quadratic function. It is expressed as f(x) = ax^{2} + bx + c, where a â‰ 0 and a, b, c are constant & x is a variable.Â The domain and the range are **R.Â **The graphical representation of a quadratic function say, f(x) = x^{2} – 4 is

- Cubic Function: A cubic polynomial function is a polynomial of degree three and can be denoted by f(x) = ax
^{3}+ bx^{2}+ cx +d, where a â‰ 0 and a, b, c, and d are constant & x is a variable. Graph for f(x) = y = x^{3}â€“ 5.Â The domain and the range are**R.**

### Rational Function

A rational function is any function which can be represented by a rational fraction say, f(x)/g(x) in which numerator, f(x) and denominator, g(x) are polynomial functions of x, where g(x)Â â‰ 0.Â Let a function f: **R**Â â†’ **R** is defined say, f(x) = 1/(x + 2.5). The domain and the range areÂ **R**. The Graphical representation shows asymptotes, the curves which seem to touch the axes-lines.

### Modulus Function

The absolute value of any number, c is represented in the form of |c|.Â If any function f: **Râ†’**Â **R** is defined by f(x) = |x|, it is known as Modulus Function. For each non-negative value of x, f(x) = x and for each negative value of x, f(x) = -x, i.e.,

f(x) = {x, if x â‰¥ 0; – x, if x < 0.

Its graph is given as, where the domain and the range are **R.**

### Signum Function

A function f: **Râ†’**Â **R** defined by

f(x) = { 1, if x > 0;Â 0, if x = 0;Â -1, if x < 0

Signum or the sign function extracts the sign of the real number and is also known as step function.

### Greatest Integer Function

If a function f: **Râ†’**Â **R **is defined by f(x) = [x], x âˆˆ X. It round-off to the real number to the integer less than the number. Suppose, the given interval is in the form of (k, k+1), the value of greatest integer function is k which is an integer. For example: [-21] = 21, [5.12] = 5. The graphical representation is

## Solved Example for You

**Question 1: Which of the following is a function?**

**Answer :** Figure (iii) is an example of a function. Since the given functionÂ maps every element of A with that of B. In figure (ii),Â the given functionÂ maps one element of A with two elements of B (one to many). Figure (i) is a violation of the definition of the function. The given function does not map every element of A.

**Question 2: What is meant by function and what are its types?**

**Answer:** A function refers to a special relation which maps each element of one set with only one element belonging to another set. The various types of functions are as follows:

- Many to one function
- One to one function
- Onto function
- One and onto function
- Constant function
- Identity function
- Quadratic function
- Polynomial function
- Modulus function
- Rational function
- Signum function
- Greatest integer function

**Question 3: How does the working of a function take place?**

**Answer**: A function refers to an equation that provides only one answer for y for every x. There is an assignment of exactly one output to each input of a specified type by a function.

**Question 4: How can one determine if a function is a function?**

**Answer:** One can determine whether an equation is a function by solving for y. In case of an equation and a specific value for x, there shall be only one corresponding y-value for that particular x-value.

**Question 5: Can we say that a parabola is a function?**

**Answer:** Only those parabolas which open downwards or upwards are functions.

The example for onto function doesn’t qualify as a function in the first place. Does it??

It is a relation but not a function because a single element in the domain has been mapped to two elements in the co domain. Isn’t it??

Please tell me if I’m correct or not.

It is really confusing.

An onto function exists if and only the co-domain is equal to the range that is every element in set A (the domain) is mapped to every element in set B (the range/codomain) i.e without leaving out any element. Irrespective of whether it is a one to one mapping or not. Therefore it is a function. Put simply, take set A as a set of sons and set B as a set of fathers, a function requires that every son has one father (which is normal) yet every father can have more than one son(which is also normal) so if one element in set A maps to more than one element in set B it is not a function (and we will need DNA test to know who really is the father) but if more than one element in set A maps to one element in set B it is still a function (the elements are just brothers). What and onto function requires is that every father has a son. Sorry if I made it a bit complicated I feel that if I continue I might make it worse just it study a bit more and from different sources, videos or books and you will understand it better.