Introduction to Functions

A Detailed Overview of Calculus, Which is Never Heard Before !
Written by: Robert Pinterson

Earned my Ph.D. in mathematics from University of North Carolina at Chapel Hill.
I am a lecturer with over 5 years of teaching experience and an active researcher in the field of quantum information. Passionate about everything connected with maths in particular and science in general.

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Concept of Variable, Constant, Expression, Cofficient

Calculus

Differential Calculus

Functions:

It is a special type of relationship in which we give values as input and which gives result as output.

Example:

$$B=x^3$$

or

$$y=f(x)$$

Function should have Specific Name: A function has a specific name and also some function has no name then we have to give name(as f and many other of your choice) to them to make function useable.

Example of function with no name:

$$B=x^3$$

Parts of function

  1. The input
  2. The relationship
  3. The output

Function Representation:

Function Representation

How to read Functions

$$y=f(x)$$

Here we read function as y as a function of “x”

Here y is use as dependent variable(depend on x) and x is independent variable.

When we give value to x (input) we have answer in y(output).

Formal Definition:

A function basically describe relation (one and only one) between elements of two sets

$$F(x)= x^3$$

function of cube

$$F(1)=1$$

mean putting value x in function as we put x as input we have output 1

$$F(2)=8$$

Function Representation

Let

$$F:X \rightarrow Y$$

is a function from set X to Y here the Set X is called Domain and Set Y is called Range of a function

So, Range of

$$f(x)={1,2}$$ $$Domain of f(x)={1,8}$$

Types of Function

  1. Algebraic Function: $$y=x2+1$$
  2. Trigonometric Functions:

    y = sinx or cosx or tanx or secx or cotx or cosecx

    graph

  3. Exponential Functions:

    $$y=e^2x$$
  4. Logarithmic Functions

    $$y=a^x$$

    Then,

    $$x=log Y a$$

    where a>0

  5. Explicit Functions:

    When we express y as function of “x”

    $$x^2+1-y=0$$ $$y=x^2+1$$
  6. Implicit Functions:

    When y cannot be expression as function of “x”

    $$x^2+1-xy=5$$