## 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.

## Explain the Types of Algebraic Functions with Examples Brainly

some more types of functions are:

### 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:

### 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:

### 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 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.

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.

f(x) = 2x

f(x) = 9x +4

### Explain Function and its Types in Math

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

## 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.

## 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.

### 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, +∞)

### 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

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