The relationship defined between one independent variable to another dependent variable is called a **function**. It basically relates an input to an output. The relationship is commonly denoted as *y=f(x)*, which in words is said “f of x”. In a mathematical sense, a **function** is a method or a relationship that connects each member *‘a’ *of a non-empty set A to at least one element *‘b’ *of another non-empty set B. In arithmetic, 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).Therefore a **function** is denoted as *f = {(a,b)| for all a ∈ A, b ∈ B}. *

## Types of **Function** based on Mapping of Elements

We have four **functions** that are based on the element mapping from set A to set B.

- If the pictures of separate components of A under
*f*are distinct, i.e., for every*a*,*b*in A,*f(a) = f(b)*,*a = b,*then*f: A→B*is said to be one-to-one or injective. Otherwise, it’s a one-to-many situation. *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. there is an element a in A such that*f(a) = b*for every*b ∈ B*. If and only if the**function’s**range equals B, the**function**is onto.- If
*f*is both one-one and onto, it is said to be one-to-one and onto or bijective.

### Vertical Line Test of a **Function**

The term “single-valued” refers to the fact that no vertical line on a graph ever crosses more than one value. It is still a legitimate curve, but not a **function** if it crosses more than once. The vertical line test determines whether or not a curve is a **function**. The curve is not a **function** if it cuts a vertical line at more than one point.

#### Types of **Function** on the basis of Graphing

- A polynomial
**function**can be expressed as y=f(a)=f0+f1a+…+fnan, where n*∈ N.*The graph is always a straight line. - Linear
**functions**are of the form ax+b=0 where a,b*∈ R,*and a are not equal to zero. The graph of such a**function**will be a straight line. - Any two
**functions**are called identical**functions**if the domain of the first**function**is equal to the domain of the second**function**and the range of the first**function**is also equal to the range of the first function. - Quadratic
**Functions**can be expressed as y=ax2+bx+c amd a,b,c*∈*R, a is not equal to zero. The graph will be a parabola. - Rational
**Function**are**functions**of the type f(a)g(a) and g(a) is not equal to zero. Graph of such**functions**is an Asymptote. - An algebraic
**function**is a function with a finite number of terms including powers and roots of the independent variable x and fundamental operations like addition, subtraction, multiplication, and division. - Cubic
**Function**is the polynomial function of degree 3. - Modulus
**Function**is defined as f(a)=|a|=a, a≥0 and f(a)= -a, a<0. - Signum
**Function**is also called the sign**function**, which gives the sign of real number. The domain is the where the**function**is defined and range is always {-1,0,1} - Greatest Integer
**Function**is defined as f(a)=[a] gives the value of the greatest integer less than or equal to a. It is always an integral output. Its domain is R and Range is integers. - Fractional Part
**Function**gives fractional value as an output. - Even an Odd Function– If f(x) = f(-x), the
**function**is an even one, and if f(x) = -f(-x), the**function**is an odd one. - If there exists a positive real integer T such that f(u – t) = f(x) for all x
*∈*Domain, the**function**is said to be a periodic**function**. - Composite
**Function**– Let f and g be two**functions**, then the composition of f and g is denoted as f(g) and is defined as fog=f(g(x)). - Constant
**Function**is the polynomial function with degree 0. The graph is a straight line parallel to x-axis. - Identity
**Function**is defined as b=f(a)=a. The graph of such a**function**is a straight line passing through the origin.

### Algebra of **Functions**

The algebra of **functions** is concerned with **function** operations. We have the following for the **functions** f(x) and g(x), where f: X*→* R and g: X *→*R, respectively, and x*∈ *X:

- (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)), where k is any real number
- (f/g)(x)=f(x)/g(x), where g(x) cannot be equal to zero.

**Conclusion**

A **function** is a mathematical formula that connects inputs and outputs. A **function** relates elements in a set (the domain) to elements in another set (the codomain). The range refers to all outputs (the actual values associated with them). A **function** is a form of relationship in which the domain’s elements are all included, and there is just one output for each input (not this or that). An ordered pair is made up of an input and its corresponding output. As a result, a **function** may alternatively be thought of as a collection of ordered pairs.