What are the Different Types of Functions? Definition & Examples

What do you Mean by Function or algebraic functions?

A function f from a set X to a set Y is a rule or correspondence that assigns to each element x in X a unique element y in Y.

Define the Range of a Function?


The set of corresponding elements y in Y is called the Range of a function.

How to Define the Domain of a Function?


The set X is called the Domain of a function.

What is the Notation and Value of a Function?


if any variable y depends on any variable x in such a form that each value of a function determines exactly one value of that y function, then it is said to be y is the function of x.


What are the Different Types of Functions? Definition & Examples


Explain the Types of Algebraic Functions with Examples Brainly


some more types of functions are:

1Algebraic functions
2- Trigonometric functions
3- Inverse Trigonometric functions
4- Exponential function
5- logarithmic function
6- Hyperbolic function
7- Inverse Hyperbolic function
8- Explicit function
9- Implicit function
10- parametric functions
11- Even function
12- Odd function


1- What is Meant by Algebraic Functions and Types


A function is said to be an Algebraic function if it is defined by some algebraic expressions.

Explain the Types of Algebraic Functions with Examples Brainly

some important types of Algebraicnfunctions are:

(a) polynomial function
(b) Linear function
(c) Identity function
(d) Constant function
(e) Rational function

What is polynomial function and example?

A function p\left(x\right)\;=\;a_nx^{n\;}+\;a_{n-1}x^{n-1}+\;a_{n-2}x^{n-2}+…+\;a_0
is said to be polynomial function if All coefficients i.e,a_{n\;},\;a_{n-1} are all real Numbers and its exponents are all non_negative (positive) integers.
its domain and range are subsets of real numbers.

Example of a polynomial function


3x^4\;+\;4x^2\;+\;6
Note that its degree is 4 with a leading coefficient of 3.

Define Linear function explain with Examples?

if the degree of any polynomial function is 1, then it is said to be a Linear function. Note that its main condition is that the degree should be 1.

Example of a Linear Function in Simple way

f(x)= ax+b (a≠0), a and b can be any real numbers.
f(x)=3x+7
its domain and range are set of real numbers.

What is Identity function give a Example


For any set X a function I:X\rightarrow X of the form I\left(x\right)=x\forall x€X it is said to be an Identity function.

what is domain and range of Identity function?


Domain and range of Identity function are set X itself.
in simple manners:
A function is said to be identity if we add a multiplicative identity element that is 0, in any function or add an additive identity element that is 0 in any element, its answer will remain the same or function will remain the same.

Definition of Constant Function with Examples


Let X and Y be sets of real numbers, A function C:X\rightarrow R defined by C(x)=a\forall x\in X\;,\;a\in Y and it is fixed, it is said to be A constant function.


Example of constant function


C:\mathbb{R}\rightarrow\mathbb{R} defined by C(x)=2\;\forall\;x\in\;R is the simple example of constant function.

Rational Function Examples with Answers


A function in the form of \frac{P(x)}{Q(x)}, if P(x) and Q(x) are the polynomial functions and Q(x)≠0, then it is said to be a rational function.

Domain and Range of Trigonometric Functions Examples

Trigonometric functions are denoted as:

Trigonometric Functions Domain Range
y=sinx R-1≤y≤1
y=cos x R-1≤y≤1
y=tan x {x:x∈R and x≠(2n+1) π\2, n an integer} R
y=cot x{x:x∈R and x≠nπ, n is any integer} R
y=sec x {x:x∈R and x≠(2n+1)π\2, n is any integer} y≥1, y≤-1
y=Csc x {x:x∈R and x≠nπ, n is any integer} y≥1, y≤-1

How to Inverse a Trigonometric Functions

inverse Trigonometric functions are denoted as:
Y= Sin^{-1}X\;\leftrightarrow\;X=Siny where \frac{-\pi}2\leq y\leq\frac\pi2,\;-1\leq x\leq1
y=cos–¹x⇄(right left double arrow)x= cos y, where 0≤y≤π, -1≤x≤1
y= tan–¹x⇄(right left double arrow)x=tan y, where -π\2

How to Exponential a Trigonometric functions

A function, in which the variable appears in exponent form (power), it is said to be exponential function.


Example of Exponential a Trigonometric functions


y=e^ax
y=2^x

            5 logarithmic function

if x=a^y, then y=Log (base a) x, where a>0 and a≠1
it is said to be logarithmic function.

         6 Hyperbolic function

sinhx=1/2(e^x – e^-x) is called hyperbolic sine function.
coshx=1/2(e^x + e^-x) is called hyperbolic cosine function.
tanhx=(e^x – e^-x)(e^x + e^-x) is called hyperbolic tangent function.
sechx=(e^x + e^-x)/2 is called hyperbolic secent function.
Cosec hx=(e^x – e^-x)/2 is called hyperbolic cosec function.
cothx=(e^x + e^-x)(e^x – e^-x) is called hyperbolic tangent function.

            7 Inverse hyperbolic function

following are Inverse hyperbolic function:
sinh–¹ x= ln(x+√x²+1)
cosh–¹x=ln(x+√x²-1)
tanh–¹ x=1/2 ln(1+x/1-x),|x|<1
coth–¹x=1/2 ln(x+1/x-1),|x|<1
sech–¹x=ln[1/x+(√1-x²)/x]
cosech–¹x=ln[1/x+(√1+x²)/|x|]

            8 Explicit function

if any y variable is easily expressed in the terms of independent variable x, then y is called an explicit function.
its simple example is Y=x²+3x-3

             9 implicit function

if x and y are so mixed up and y cannot be expressed in the terms of independent variable x, then y is called an implicit function of x.
its simple example is x²+xy+y²=3

              10 parametric functions

The functions of the form:
x=at² & y=at
x=a cos t & y=a sin t
x=a cos theta & y=b sin theta
x=a sec theta & y=a tan theta
are said to be parametric functions.
here t and theta are called parameters.

        11 Even function

A function f is said to be an Even function if
f(-x)=f(x), for every Number x in the domain of f.
its most common example is cos(-x)=cos(x)=f(x). cos is an even function.

           12 Odd function

A function f is said to be an Even function if
f(-x)=-f(x), for every Number x in the domain of f.
Its most common example is sin(-x)=-sin x =-f(x)

What is the Difference Between Into and Onto Function

What is the Difference Between Into and Onto Function

What is the Best Definition of Function in Math

A special type of relation is a function defined as below:

  • f is a relation from A to B that is, f is a subset of A × B.
  • Dom f= A.
  • First element of no two pairs of f are equal, then f is said to be a function from A to B.


A function is also written as


f: A➝B. Which is read: f is a function from A to B.

What is the Best Definition of Function in Math


Every function is a relation. but every relation is not a function.


Define Function with Details In simple words


if A and B are two sets then relation f: A➝B is called a function if for every element of A there exists a unique integer in B and Dom f=A. its main identification is When the domain does not repeat, it is a function.


What is the Best Examples of a Function

f(x) = 2x

f(x) = 9x +4

Explain Function and its Types in Math

There are various types of functions. Some are given below:

Into function
Onto Function ( Surjective Function )
(1_1) and Into Function (Injective Function )
(1_1) and Onto Function ( Bijective Function )
Linear Function
Quadratic Function
Fun types

Difference Between Into and Onto Function

What does Into mean Function in Math?

If a function f: A➝B is such that Ran f⊂B i.e., Ran f≠B, then f is said to be a function from A into B.
In simple words:
A function is said to be into a function if every element of A is busy with elements of B.

What does Onto mean in Math

if a function f: A➝B is such that Ran f=B i.e., every element of B is the image of some elements of A, then f is called an onto function. It is also called the surjective function.
in simple words:
A function is said to be onto function if every element of B is also busy with elements of A.

What is (1_1) and INTO (Injective) Function?

if a function f from A into B is such that the second element of no two of its ordered pair are equal, then it is called an injective (1_1) and into function.
in simple words:
A function is said to be (1_1) and into an (injective) function if in B there is at least one nonbusy element.

What is (1 _ 1) and ONTO Function (bijective function)?

If f is a function from A onto B such that the second element of no two of its ordered pairs are the same, then f is said to be (1 1) function from A onto B. Such functions are also called (1 1) correspondence between A and B. It is also called bijective function.
In simple words: A function is said to be (1 _ 1) and onto function (bijective function) if every element of A is busy with every element of B.

What does Linear Function Definition in Math?

The function {(x,y) | y= mx + c} is called a linear function. Because its graph is a straight line.

Define Quadratic Function in Math?

y = mx + c or ax + by + c= 0 represents a straight line. This can be easily verified by drawing graphs of a few linear equations with numerical coefficients. The function { (x , y) | ax² + bx + c} is called a quadratic function because it is defined by a quadratic (second degree) equation in x , y.

Explain Limits of Functions Examples and Definition

Explain Limits of Functions Examples and Definition

Function and Limits

Function are vital gear by way of which we describe the real world in mathematical phrases. They may be used to give an explanation for the relationship among variable quantities and hence play a crucial function within the study of calculus.

Define Limit of a Function in Calculus

A Function f from a set X to a set Y is a rule or a correspondence that assigns to each element x in X a completely unique element y in Y. The set X is known as Domain of f.

The set of corresponding elements y in Y is known as the Range of f. Also, except stated to the contrary, we will count on hereafter that the set x and y consist of actual numbers.

What is the Real Concept of Function?

The term Function was recognized by a German Mathematician Leibniz (1646-1716) to describe the dependence of one quantity on another. The subsequent examples show how this term is used:

  • The area “A” of a square depends on one of its sides “x” by the formula A = \boldsymbol x^{\mathbf2} , so we say that A is function of x.
  • The volume “V” of a sphere depends on its radius “r” by the formula V = \frac{\mathbf3}{\mathbf4}\boldsymbol\pi\boldsymbol r^{\mathbf3} , So we say that V is a Function of r.

A feature is a rule or correspondence referring to two units in this sort of manner that every element inside the first set corresponds to at least one and simplest one detail inside the

  • Thus in (1) above, a square of a given side has only one area.
  • And in (2) above, a sphere of a given radius has only one volume.

Notation and Value of Function

If a variable y depends on a variable x in such a way that each of determines exactly one value f y, then we says that Y is a function.

Note for Function

Note: Functions are generally denoted by some letters such as f, g, h, F, G, H, and so on.
Notation and Value of Function

A Function can be thought as a computing machine f that takes an input x, operate on it in some way, and produces exactly one output f(x). This output f(x) is called the value of f at x or image of x under f. The output f(x) is denoted by a single letter, say y, and we write y = f(x).

Dependent and Independent Variable in Function

The Variable x is called the independent variable of f, and the variable y is called the dependent variable of f. for now onward we shall only consider the function in which the variable are real numbers and we say that f is real valued function of real numbers.