# Functions and its Types

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## Types of Functions

The true meaning of functions is that they are the relations at the place of which each input has a specific output.

‘Functions and its types’ chapter explains the basic concepts, as well as, the various types of function in math. All are described below using several examples for better understanding.

The very definition of a function is: it is an association between a set of inputs and a set of allowable outputs with the ability that each input is linked to exactly one output.

Here you can learn about various types of functions, as well as the types of functions graphs, in mathematics, in brief.

### Types of Functions in Mathematics

The types of functions in Mathematics are:

• One to One Function

• Many to One Function

• Onto Function

• One - One and Onto Function

• Identity Function

• Constant Function

• Polynomial Function

• Rational Function

• Modulus Function

• Signum Function

• Greatest Integer Function

### Different Types of Functions

The different types of functions explain various mathematical relations and their graphical representation.

1. Identity Function

Consider R as the set of real numbers. The function is said to be the Identity function, if f: R→R is explained as f(x) = y = x, for x ∈ R, the domain and the range being R.  The types of functions in sets can be understood with the help of these functions.

The Identity Function graph is a straight line and always passes through the origin.

1. Constant Function

Here, the condition about the function f: R→R is distinct as f(x) = y = c, for x ∈ R and c is a constant in R, at this point such a function is identified as Constant function. R is The domain of the function (f), and its range is a constant (c).

By plotting the graph, we discover a straight line parallel to the x-axis.

1. Polynomial Function

A polynomial function is explained by y = a+ a1x + a2x2 + … + anxn, here ‘n’ is a non-negative integer and a0, a1, a2,…, n ∈ R. The uppermost power in the equation is the degree of the polynomial function.

Also, these polynomial functions are categorized as per their degrees:

• Constant function: The polynomial function will be constant if the degree is zero.

• Linear function: This kind polynomial function possesses the degree ‘one’. For instance;

y = x + 1 or y = x or y = 2x – 5, etc.

If we consider the equation, y = x – 6. R is the domain as well as the range. The graph is a straight line.

A function is said to be a quadratic function if the degree of the polynomial function is two. This can be represented as f(x) = ax2 + bx + c, where a ≠ 0 and a, b, c is constant, and x is a variable. R is the domain and the range. The plotting in graph of a quadratic function for the function: f(x) = x2 – 4 is

1. Rational Function

A rational function can be characterized such as, f(x) / g(x) where both the f(x) and g(x) are polynomial functions of x, at a condition g(x) ≠ 0.

If f: R → R and it says that f(x) = 1 / (x + 2.5).

The Graphical representation shows asymptotes, the curves which seem to touch the axes-lines.

1. Modulus Function

The outright value of any number, c is characterized in the form of |c|. It can be said that f: R→ R is well-defined by f(x) = |x|.

f(x) = x: for each non-negative value of x, and

f(x) = - x: for each negative value of x

For example

f(x) = {- x, if x < 0

= {– x, if x ≥ 0.

1. Signum Function

f: R→ R can be explained as

f(x) = {0, if x = 0

= {1, when x > 0

= {-1, if x < 0

1. Greatest Integer Function

The condition for a function f: R→ R is denoted by f(x) = [x]; such that x ∈ X. It completes the real number to the integer less than the number. The types of functions in sets become easier to comprehend with the help of respective graphs.

Assume that the interval is in the shape of (k, k+1), the utmost integer value of the function is k, which is an integer.

For illustration:

[-21] = 21, [5.12] = 5. The graphical demonstration is

### Types of Function Graphs

Here you can learn about the different types of functions and their graphs easily by referring to the illustrations.

Some common types of function graphs are shown below.

### Types of Functions in Sets

There are various types of functions that are propagated in sets. These  types of functions in sets are discussed below.

i. One-to-One Function

The condition for a function f: A → B to be One to One when each element of A is an individual element of B.

ii. Many-to-One Function

This kind of function plots two or more elements of A to the identity element of set B.

iii. Onto Function

A function is called onto function, which represents that each element of set B is (are) preimage(s) in set A.

iv. One - One and Onto Function

This function is possible when each element of A is a distinctive element of B.

Q1: Explain the working procedure of a Function.

Ans: A function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set. Otherwise, a function denotes an equation that delivers only one answer for ‘y’ for every value of ‘x’.

This is the best chance to get the precise one output to a respective input of an identified type by a function.

Q2: How can one determine if an equation is a Function?

Ans: One can determine whether an equation is a function by solving for y.

When you solve an equation with a specific value for x, there shall be only one corresponding y-value for that particular x-value.

Q3: Can you identify a Function as given in the following figures?