# Functions

In mathematics, functions are the most fundamental aspect of calculus. Functions are certain forms of relationships. A function is represented as a rule that produces a unique output for each input x.

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 xX:

• (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.

Get answers to the most common queries related to the CA Foundation Examination Preparation.

## What is the basic concept of function in math?

Ans. If every element of set A has exactly one image in set B, the relation is...Read full

## What are the parts of the functions?

Ans. Domain, range, and co-domain are the three core ideas that help define an...Read full

## What are some of the examples of Functions?

Ans. Depending on how much money you put...Read full

## What are the different transformations of functions?

Ans. The different transformations are reflection, translation, rotation, and dilation.

## How do you prove that a function is onto?

Ans.  A function is onto if and only if the range is equal to the co-d...Read full