**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:**

1– Algebraic 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 Algebraic*n

**functions 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)