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)

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