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.

How we Define Hyperbolic Functions and their Simple Formulas

Hyperbolic Functions

Define Hyperbolic Function in Simple Way

  • Sin Hyperbolic Function
  • Cos Hyperbolic Function

Simply Define sinh x hyperbolic function

Sinh x = \frac12\left(e^x-\;e^{-x}\right) is called hyperbolic sine function. Its domain and range are the set of all real numbers.

Define sinh x hyperbolic function

Formula for Cosine hyperbolic function in Math with Detail

Cosh x = \frac12\left(e^x+\;e^{-x}\right) is called hyperbolic cosine function. its domain is the set of all real numbers and the range is the set of all numbers in the interval.

[1, +∞)

Formula for Cosine hyperbolic function

Four Hyperbolic Functions and Formulas and Their Details

The remaining four hyperbolic functions are defined in the terms of the hyperbolic sine and the hyperbolic cosine function as follows:

Four Hyperbolic Function Formulas are

Functionsis equal toFormulas
tanh x\frac{\sin h\;x}{\cos h\;x}\frac{e^x\;-\;e^{-x}}{e^x\;+\;e^{-x}}
sech x\frac1{\cos h\;x}\frac2{e^x\;+\;e^{-x}}
coth x\frac{\cos h\;x}{\sin h\;x}\frac{e^x\;+\;e^{-x}}{e^x\;+\;e^{-x}}
csch x\frac1{\sin h\;x}\frac2{e^x\;-\;e^{-x}}

The hyperbolic functions have the same properties that resemble to those of trigonometric functions.

Method to Draw the Graphs of Algebraic Functions

Defined Graphs of Algebraic Functions

If f is a real-valued function of a real numbers, then the graph of f in the xy-plane is defined to be the graph of equation y = f(x). The graph of a function f is the set of point { (x, y)| y = f(x), x is in thedomain of f } in the cartesian plan for which (x, y) is an ordered pair of f.

The graph provides a visual technique for determining whether the set of points represents a function or not. if a vertical line intersects a graph in more than one point, it is not the graph of a function.

Explain is given in the figure.

Sketch the Graph of a function

A function Graph

Defined Graphs of Algebraic Functions
A Function

Is Also a Function Graph

Sketch the Graph of a function

The Sketch is not a function

The Sketch is not a function
not a function

Best Method to Draw the graph

To draw the graph of y = f(x), we give arbitrary value of our choice to x and find the corresponding values of y. In this way we get ordered pairs \left(x_1\;,\;y_1\right) , \left(x_2\;,\;y_2\right) , \left(x_3\;,\;y_3\right) etc. These orders pairs represent points of the graph in the cartesian. We have these points and join them together to get the graph of the function.