Learn About Functions and Its Types with Examples

Functions are relations in which each input has a specific output. Let’s learn what functions in mathematics mean and the different types of functions using various examples for better understanding.

## What Are Functions?

A function is a process or a relation that connects each element ‘a’ of a non-empty set A to at least one additional non-empty set B. In mathematics, a function is a relation f from one set A (the domain of the function) to another set B (the co-domain of the function). For any a∈A, b∈B, f =(a,b).

If every element of set A has exactly one image in set B, the relation is said to be a function.

A function is a relation from a non-empty set B with the domain A and no two separate ordered pairs in f having the same first element.

A function from A → B and (a,b) ∈ f, then f(a) = b, where ‘b’ is the image of ‘a’ under ‘f’ and ‘a’ is the preimage of ‘b’ under ‘f’.

If there is a function f: A → B, set A is termed the function f’s domain, and set B is called its co-domain.

**Types of Functions**

In mathematics, there are numerous types of functions. Let’s explore each of them in detail.**Modulus Function**

A modulus function is a type of function that determines the absolute value of a number by its magnitude. It is written as f(x) = |x|. It is further defined as follows:

f(x)={x or-x}, where x≥0 or x<0 respectively.**One – one function (Injective function)**

If each element in a function’s domain has a distinct image in the co-domain, the function is said to be one-one.

**For examples:**

f: R R given by f(x) = 3x + 5 is a one-one function.

Many – one function

If there are at least two elements in the domain with the same image, the function is known as many to one.

**For example:**

f : R R given by f(x) = x2 + 1 is a many one function.

Onto – function (Surjective Function)

An onto function is one that has at least one pre-image in the domain for each element in the co-domain.**Identity Function**

The identity function is a function that has the same input as it does output. It is written as f(x) = x, where x ϵ R. f(3) = 3 is an example of an identity function.**Polynomial Function**

Polynomial functions are functions that can be expressed as polynomials. It is written as,

f(x)=anxn+an−1xn−1+…….a0x0.

For example, f(x) = 2x+3 or f(x)= x3 – 4×2+ 13x – 6 is a polynomial function.**Linear Function**The polynomial function with degree as one. For example, y = x + 1 or y = x or y = 2x – 5 and so on. Taking this into account, y = x – 6. R is the domain as well as the range.

**Constant Function**

A constant function is one that always returns the same value as input. The formula is f(x) = c, where c is a constant. An example of constant function is f(x) = 2.

**Quadratic Function**

A quadratic function is one of the polynomial functions with the highest power 2. It is written as, f(x)=a(x-h)2+k.

f(x) = x2 + 3, for example, is a quadratic function.**Cubic Function**

A cubic function is a type of polynomial function that has degree 3. It is written as, f(x) = ax3 + bx2 + cx + d, where a, b, c, d are constants.

f(x) = x3 + 5, for example, is a cubic function.**Rational Function**

A Rational function is a function that is derived by dividing two polynomial functions by their ratio. It is written as, f(x) = P(x)/Q(x), where P and Q are polynomial functions of x and Q(x) is zero.

f(x)=x2+2x+1×2−4, for example, is a rational function.