Functions are relations where each input has a specific output. This lesson covers the concepts of functions in mathematics and the different types of functions using various examples for better understanding.

**Feature Related Content**

- functions
- Borders, continuity and differentiability
- differentiation
- derivative applications

## JEE Main 2021 Maths LIVE Paper Solutions 24. Feb Turn-1 Based on memory

## What are functions in mathematics?

A function is a relation between a set of inputs and a set of allowed outputs with the property that each input is associated with exactly one output. Let A and B be any two non-empty sets; The mapping from A to B is a function only if every element in set A has an edge, only one image in set B.

**Example:**

Another definition of a function is that it is a relation "f" in which each element of the set "A" is associated with a single element belonging to the set "B". Also in a function there cannot be two pairs with the same first element.

### A condition for a function:

Define**A**and adjust**B**must not be empty.

In a function, a specific input is given to get a specific output. So a function**f: A->B**means that f is a function of**A**Pro**B**, Where from**A**is a domain and**B**is a codomain.

- per article
**a**who listened**A**, a∈ A,a single article**b**, b∈Bexists such that (a,b)∈F.

the unique article**b**which**F**Regards**a**, is denoted by**Fa)**and it's called**f of one**, or the value of**f in one**, o**the image of a bass f.**

- o
*reach*Von**F**(Picture of**a**sollozo f) - is the set of all values of
**f(x)**taken together. **area f**= {j∈AND | y = f(x), for some x in X}

has a real valued function**PAG**or one of its subsets as its domain. Even if your domain is too**PAG**or a subset of it**PAG**, dice**actual function**.

**Vertical line test:**

The vertical line test is used to determine whether or not a curve is a function. If a curve intersects a vertical line at more than one point, the curve is not a function.

### function representation

Functions are usually represented asf(x).

you are f(x) = x^{3}.

We say that f of x is equal to x cubes.

Functions can also be represented by g(), t(),...etc.

### Steps to solve functions

**Question:**Find the output of the function g(t) = 6t^{2}+ 5 Em

(yo) t = 0

(ii) t = 2

**Solution:**

The given function is g(t) = 6t^{2}+5

(i) At t = 0, g(0) = 6(0)^{2}+ 5 = 5

(ii) Em t = 2, g(2) = 6(2)^{2}+ 5 = 29

## types of functions

There are different types of functions in mathematics, which are explained in more detail below. The different types of functions covered here are:

- One - a function (injective function)
- many - one role
- in function (surjective function)
- function
- polynomial function
- lineal funtion
- identical function
- quadratic function
- rational function
- algebraic functions
- cubic function
- module function
- signal function
- largest integer function
- fraction function
- even and odd function
- periodic function
- compound function
- constant function
- identity function

**Elaborate:**Find the missing equations in the graphs above.

## Features - Video Lessons

## functions and types of functions

## number of functions

## even and odd functions

## Composite and periodic functions

### One - a function (injective function)

If each element in the scope of a function has a specific image in the scope, the function is calledone - one function.

**For example**F; R R given by f(x) = 3x + 5 is one – one.

### many - one role

On the other hand, if there are at least two elements in the domain whose images are equal, the function is said to be a many-to-one function.

**For example**f : R R dada por f(x) = x^{2}+ 1 is many one.

### in function (surjective function)

A function is calledin functionif each element in the range has at least one ancestor in the domain.

### function

If there is at least one element in the range that is not an image of an element in the domain, the function is Into.

(Q) Let A = {x : 1 < x < 1} = B be a function f : A B, find the nature of the given function (P). F(x) = |x|

f(x) = |1|

Solution for x = 1 & -1

Therefore, it is very important that the range of f(x) be from [-1, 1] to [0, 1], which is not equal to the codomain.

So it works.

Suppose we have a function,

\(\begin{array}{l}f(x)=\left\{\begin{matriz} x^2 & ; & x\geq 0\\ -x^2 & ; & x<0 \end{matriz }\direita.\end{matriz} \)

For different input values, we have different outputs, so it's... a function that is also handled equals its codomain, so that's it too.

### polynomial function

A real valued function**f : PAG → PAG**defined by

\(\begin{matriz}{l}y = f(a) = h_{0}+h_{1}a+…..+h_{n}a^{n}\end{matriz} \)

, Where from**norte∈ norte mi h**it's called p

_{0}+h_{1}+ … + h_{norte}∈ P, for each**a ∈ P**,**polynomial function.**

**norte**= a non-negative integer.- o
**degree**of the polynomial function is the highest power in the expression. - yes or
**the degree is****Null**, is called a constant function. - yes or
**grade is in**and is called a linear function. Example: b = a+1. - Graph type: Always a straight line.

Therefore, a polynomial function can be expressed as:

\(\begin{matriz}{l}f(x)= a_{n}x^{n}+a_{n-1}x^{n-1}+…..+a_{1}x^{ 1}+a_{0}\end{matriz} \)

The highest power in expression is known as**Degree of the polynomial function**. The different types ofpolynomial functionsby grade are:

- The polynomial function is called a constant function if the degree is zero.
- The polynomial function is called linear if the degree is one.
- The polynomial function is quadratic if the degree is two.
- The polynomial function is cubic if the degree is three.

### lineal funtion

All functions in the form of ax + b, where a, b∈R& a ≠ 0 calledlinear functions. The graph will be a straight line. In other words, the linear polynomial function is a first degree polynomial where the input must be multiplied by m and added to c. It can be expressed by f(x) = mx + c.

Example: f(x) = 2x + 1 at x = 1

f(1) = 2,1 + 1 = 3

f(1) = 3

Another example of a linear function is sy = x + 3

### identical function

Of the**The functions f and g must be identical**what if

**(a)**The domain of f = domain of g

**(b)**The area of f = the area of g

**(C)**f(x) = g(x)∀ x ∈ D_{F}&D_{gram}

**For example**f(x) = x

\(\begin{matrix}{l}g(x) = \frac{1}{1/x}\end{matrix} \)

Solution: f(x) = x is defined for all x

but

\(\begin{matrix}{l}g(x) = \frac{1}{1/x}\end{matrix} \)

is undefined of x = 0So it is identical for x∈ R-{0}

### quadratic function

All functions in the form y = ax^{2}+ bx + conde a, b, c∈R, a ≠ 0 will be known as a quadratic function. The graph will be parabolic.

\(\begin{matriz}{l}\text{At}\ x=\frac{-b \pm \sqrt{D}}{2}\end{matriz} \)

, we get its maximum at the minimum value depends on the leading coefficient and this value is -D/4a (where D = discriminant)in a nutshell,

A quadratic polynomial function is a polynomial of the second degree and can be expressed as;

F(x) = ax^{2}+ bx + c, and a is nonzero.

where a, b, c are constants and x is a variable.

Example, f(x) = 2x^{2}+ x - 1 y x = 2

Se x = 2, f(2) = 2,2^{2}+ 2 – 1 = 9

**For example**: y = x^{2}

**For more information, see:**quadratic function formula

### rational function

These are real functions of type

\(\begin{matriz}{l}\frac{f(a)}{g(a)}\end{matriz} \)

Where from**Fa)**mi

**Georgia)**are polynomial functions of

**a**defined in a domain where

**g(a) ≠ 0.**

- For example
**f : PAG – {– 6} → PAG**defined by\(\begin{array}{l}f(a) = \frac{f(a+1)}{g(a+2)}\forall a\in P – {-6},\end{array} \)

Is arational function. - Graph type: asymptotes (the curves touch the lines of the axes).

### algebraic functions

An algebraic equation is known as a function consisting of a finite number of terms, involving powers and roots of the independent variable x and basic operations such as addition, subtraction, multiplication, and division.

For example,

\(\begin{matriz}{l}f(x)=5x^{3}-2x^{2}+3x+6\end{matriz} \)

,\(\begin{matriz}{l}g(x)=\frac{\sqrt{3x+4}}{(x-1)^{2}}\end{matriz} \)

.### cubic function

A cubic polynomial function is a polynomial of the third degree and can be expressed as;

F(x) = ax^{3}+ bx^{2}+ cx + d y a is nonzero.

In other words, any function of the form f(x) = ax^{3}+ bx^{2}+ cx + d, onde a, b, c, d∈R& and ≠ 0

**For example:**y = x^{3}

Domain∈R

Reach∈R

### module function

the true function**f : PAG → PAG**defined by f(a) = |a| = that when that≥ 0. y f(a) = -a si a < 0 ∀ a ∈p is called*module function.*

- Domain
*Von**F**=**PAG* - reach
**F**=P^{+}tu {0}

\(\begin{matriz}{l}y=|x|=\left\{\begin{matriz} x & x\geq 0\\ -x & x<0 \end{matriz}\right.\end{ Vielfalt} \)

Domain: R

Range: [0,∞)

### signal function

the true function**f : PAG → PAG**is defined by

\(\begin{matriz}{l}\left\{\begin{matriz}\frac{\left | f(a) \right |}{f(a)}, a\neq 0 \\ 0, a= 0 \end{matriz}\right.= \left\{\begin{matriz} 1, cae x>0\\ 0, cae x=0\\ -1, cae x<0\end{matriz}\right. \end{matriz} \)

dice**signal function**o**signal function**. (indicates the sign of the real number)

- Domain of definition of f =
**PAG** - range f = {1, 0, - 1}

Example: Character(100) = 1

sign (register 1) = 0

character(x)^{2}1) =1

### largest integer function

the true function**f : PAG → PAG**defined by**f(a) = [a], a** ∈ **PAG**takes the value of the largest integer that is less than or equal to**a,**dice**largest integer function.**

- Then f(a) = [a] = – 1 for – 1 ⩽ a < 0
- f(a) = [a] = 0 for 0 ⩽ a < 1
- [a] = 1 for 1 ⩽ a < 2
- [a] = 2 for 2 ⩽ a < 3 and so on...

olargest integer functionalways gives full power. The largest integral value obtained from the input is the output.

For example: [4,5] = 4

[6,99] = 6[1,2] = 2Domain∈R

Reach∈all

### fraction function

{x} = x – [x]

It always gives a fractional value as output.

**For example**:- {4,5} = 4,5 – [4,5]

= 4,5 – 4 = 0,5

{6,99} = 6,99 – [6,99]

= 6,99 – 6 = 0,99

{7} = 7 – [7] = 7 –7 = 0

### even and odd function

If f(x) = f(-x) then the function is even & f(x) = -f(-x) then the function is odd

**Example 1:**

f(x) = x^{2}sin

f(-x) = -x^{2}sin

Here, f(x) = -f(-x)

It is a strange feature.

**Example 2:**

\(\begin{matriz}{l}f(x)={{x}^{2}}\end{matriz} \)

mi

\(\begin{matriz}{l}f(-x)={{x}^{2}}\end{matriz} \)

f(x) = f(-x)

It is an even function.

### periodic function

A function is said to be periodic if there exists a positive real number T such that f(u – t) = f(x) for every region x ε.

For example f(x) = sin x

f(x + 2π) = sen (x + 2π) = sen x fundamental

then the period of sin x is 2π

### compound function

Let A, B, C be three non-empty sets.

you are f:a→B y G: G→C can be two functions, so gof : A→C. This function is called data composition of f and g gof(x) = g(f(x)).

For example f(x) = x^{2}&g(x) = 2x

f(g(x)) = f(2x) = (2x)^{2}= 4x^{2}

g(f(x)) = g(x^{2}) = 2x^{2}

### constant function

The function**f : PAG → PAG**is defined by**second = f(x) = D**,**a** ∈**PAG**, Where from**D**is a constant∈ **PAG**, is a constant function.

- domain of
**f = pag** - reach
**f = {D}** - Graph Type: A straight line parallel to the x-axis.

In simple terms, the polynomial of degree 0, where f(x) = f(0) = a_{0}= do. Regardless of the input, the output is always a constant value. The graph for this is a horizontal line.

### identity function

**PAG**= my real numbers

The function**f : PAG → PAG**defined by**b = f(a) = un**for each∈ **PAG**is called the identity function.

- domain of
**F**=**PAG** - reach
**F**=**PAG** - Graph Type: A straight line through the origin.

__video capabilities__

__video capabilities__

### Domain, range, functional period

### roles and relationships

## Questions of relations and functions.

## One-One and Onto functions

## frequent questions

### What is a function in mathematics?

A relation f from set A to set B is called a function if every element in set A has exactly one image in set B.

### What is meant by domain of a function?

The domain of a function is the set of all possible entries of a function.

### What do you mean by image of a function?

The domain of a function is the set of all possible output values.

### What do you mean by constant function?

The constant function is a function whose output is the same for all input values. Example: f(x) = 3. Here the output is 3 for each value of x.