Table of Contents

## What is Function?

A function is a mathematical notion that connects input values to output values. It represents a rule or formula that converts inputs into outputs. Functions are denoted by symbols, such as “f(x),” and are fundamental in math, science, and computer science for modelling relationships and solving problems.

### Definition of a function

Assume A and B are sets. A function, indicated as f, is a mapping from A to B in which there exists a unique element b in B for every element a in A. The rule f(a)=b represents this mapping, where a is the input value from the domain A and b is the equivalent output value from the range B.

### Key characteristics of Function

A function’s primary properties are:

- The collection of input values for which the function is defined is referred to as the domain.
- The function’s range is the collection of output values.
- Correspondence: Each element in the domain corresponds to a distinct element in the range.
- Uniqueness: No two separate domain elements can relate to the same range element.
- Algebraic equations, diagrams, charts, and verbal explanations are all ways to express functions.
- They are widely used to represent and analyse connections, solve problems, and explain mathematical operations and transformations in many fields of mathematics, including calculus, algebra, geometry, and others.

### Several types of Functions

Functions are classified according to their qualities and characteristics. Here are some examples of popular sorts of functions:

- A
**linear function:**A linear function is one that has the formula*f(x) = mx+b*where m and b are constants. It is an example of a straight line with a constant rate of change. **Quadratic function:**A quadratic function is defined as*f(x) = ax*, where a, b, and c are constants. It denotes a parabolic curve.^{2}+ bc + c- E
**xponential function: An exponential function**is one that has the form f(x) = ae, where is a constant. It denotes either quick development or deterioration.^{x} **Trigonometric Functions:**Functions that use trigonometric ratios such as sine (sin), cosine (cos), tangent (tan), and so on. They link angles to triangle side length ratios.**Logarithmic Function:**A function of the type F(x)=loga^{x}where a is the base. It is the inverse operation of exponentiation.**Polynomial Function:**A function of the type, where the coefficients are constants.**f(x) = a**_{n}x^{m}+ a_{n-1}x^{n-1}+ a_{n-2}x^{n-2}+ … + a_{0}**Rational Function:**A function that is a ratio of two polynomial functions, such aswhere P(x) and q(x) are polynomial functions.**f(x) = p(x)/q(x)****Absolute Value Function:**A function of the typethat always returns the non-negative value of x.**f(x) = |x|****Piecewise function:**A piecewise function is one that has various rules or formulae for different intervals or specific ranges of input values.**Trigonometric Inverse Function:**Functions that reflect the inverse operations of trigonometric functions, such as arcsine (sin1), arccosine (cos1), and arctangent (tan1).

These are only a few instances of several sorts of functions. Many more specialised functions are employed in many disciplines of mathematics and other fields of study.

### Steps to check the given relation is function or not

- To determine if a given relation is a function, perform the following steps:
- Recognise the following relationship: Read and comprehend the provided relationship between the input and output values. The relationship can be expressed in a variety of ways, including a table, graph, collection of ordered pairs, or equation.
- Determine the following input and output values: Determine the domain of input values and the range of matching output values from the provided relation. It is critical to identify all potential input-output pairings.
- Check for uniqueness: For each input value, check that it corresponds to a unique output value. To put it another way, no input value should have more than one output value.
- Examine the whole domain: Check that each element in the domain has an output value in the range. Check for any “missing” or “unmapped” items.
- Check for repetition: Make certain that no output value relates to multiple input values. Each output value in the relation should be distinct.
- Use suitable representations: If the relation is presented as a table or collection of ordered pairs, inspect the values for violations of uniqueness or repetition. If a graph is supplied, ensure that no vertical lines cross more than one point.
- Make a decision: According to the study, if all the input values have distinct output values and there are no repeats or missing data,

### Examples for function

Linear Function f(x) = 2x+3. To generate the output, this function takes an input value, multiplies it by 2, and then adds 3.

The quadratic function is defined as f(x) = x2 – 4x+5 . This function includes the square of the input value , as well as linear and constant terms.

f(x)=2x is an exponential function. To compute the output, this function raises the base (2) to the power of the input value x.

f(x) = sin(x) is a trigonometric function. The sine of the input value x is computed using this function.

f(x) = x is the identity function. Without any alteration, this function just returns the input value as the output value.

## Frequently Asked Questions on Function

### What do you mean a function?

A function is a fundamental notion in mathematics that specifies a connection between two sets of items known as the domain and the range. It is a rule or correspondence that assigns each domain element to a distinct element in the range. More formally, a function f is defined as a set of ordered pairs (x,y) where x belongs to the domain and y belongs to the range, such that for every x in the domain, there exists a unique y in the range, denoted as f(x).

### Who defined function:

around for millennia, and it has been improved and polished by various mathematicians throughout history. Mathematicians such as Gottfried Wilhelm Leibniz and Leonhard Euler contributed to our contemporary knowledge of functions. Augustin-Louis Cauchy, on the other hand, was one of the most prominent mathematicians who played a vital part in developing and formalising the idea of a function. Cauchy proposed the formal definition of a function as a mathematical connection between two sets with a specified mapping between their members in the early nineteenth century.

### What is a function also called?

This function is also known as a mapping or a transformation. These phrases are frequently used interchangeably with function and denote the same mathematical notion. According to a given rule or relationship, the function translates elements from one set, called the domain, to another set, called the codomain or range.

### What is a relation and function

A relation is a broad notion that describes the relationship between items of two sets. It can be represented as an ordered pair set. A function is a form of relation in which each input has a distinct output. It specifies a predictable mapping that follows well-defined rules and characteristics. Functions are essential in mathematics and other disciplines of study.