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Function

Function

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
Representation-
Cartesian product of 2 sets is mapping every element to every element of other set. It is generally represented as X A={"a","b"}, B={"y","z"} then A X B= {("a","y"),("a","z"),(b,y),(b,z)} Relation: A Relation R from a non-empty set A to a non empty set B is a subset of cartesian product A X B. n(A)= number of element in set A. n(B)= number of element in set B. Let 's say n(A)=p, n(B)=q n(A X B)=pq then no of relations possible from A to B is 2^pq A relation f from a set A to a set B is said to function if every element of set A has one and only one image in set B. Relation: A function f is X to Y then f: X->Y X is called domain set, Y is called range function. Functions are generally represented as f(x)
Let , f(x)=x3
It is said as f of x is equal to x cube.
Functions can also be represented by g(), t(),… etc.
Lets Work Out-
Example- Find the output of the function g(t)=6t2+5 at
(i) t = 0
(ii) t = 2
Solution- The given function is g(t)=6t2+5
(i) At t = 0, g(0)=6(0)2+5
    = 5
(ii) At t = 2, g(2)=6(2)2+5
    = 29
Types of Functions-
One-to-One function – It signifies that one element of a set is related to one element of the other set.
One-to-One Function
Many-to-One-function- It signifies that many elements of a set is related to one element of the other set.
Many-to-One Function
Note :
One-to-Many- A function cannot have a one to many relation ie. one element of a set cannot be related to more than one element of the other set.
One-to-Many Function
Vertical Line Test-
Vertical line test is used to determine whether a curve is a  function or not.
If any curve cuts a vertical line at more than one points then the curve is not a function.
 Vertical Line Test
Types of Functions-

Polynomial function-

A polynomial function can be expressed as :
f(x)=anxn+an−1xn−1+…..+a1x1+a0
The highest power in the expression is known as the degree of the polynomial function. The different types of polynomial functions based on the degree are:
  1. The polynomial function is called as Constant function if the degree is zero.
  2. The polynomial function is called as Linear if the degree is one.
  3. The polynomial function is Quadratic if the degree is two.
  4. The polynomial function is Cubic if the degree is three.

Constant Polynomial Functions

The polynomial of 0th degree where f(x) = f(0) = a0=c. Regardless of the input, the output always result in constant value. The graph for this is a horizontal line.
Constant Function

Linear Polynomial Functions

A linear polynomial function is a first degree polynomial where the input needs to be multiplied by m and added to c. It can be expressed by f(x) = mx + c.
For example, f(x) = 2x + 1 at x = 1
f(1) = 2.1 + 1 = 3
f(1) = 3
Linear Function

Quadratic Polynomial Functions

A Quadratic polynomial function is a second degree polynomial and it can be expressed as;
F(x) = ax 2 + bx + c, and a is not equal to zero.
Where a, b, c are constant and x is a variable.
Example, f(x) = 2x 2 + x – 1 at x = 2
If x = 2, f(2) = 2.2 2 + 2 – 1 = 9
Quadratic Function

Cubic Polynomial Function

A cubic polynomial function is a polynomial of degree three and can be expressed as;
F(x) = ax 3 + bx 2 + cx + d and a is not equal to zero.
Cubic Function

Algebraic Functions

A function that consists of a finite number of terms involving powers and roots of independent variable x and fundamental operations such as addition, subtraction, multiplication, and division is known as an algebraic equation.
For Example-
f(x)=5x3−2x2+3x+6, g(x)=3x+4√(x−1)2..
Rational functions, irrational functions and Polynomials functions are examples of algebraic functions.

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