Differential Calculus – Functions & Graphs

Before we study differential calculus, it is important to understand the concept of functions and their graphs. This is a major pre-requisite before any Calculus course often dealt with in a separate course called Pre-Calculus.

The Concept Of Functions, Domain & Range

Lets say we have a circle. The area of this circle depends on its radius. Hence, we have an example where a quantity depends on another. Thats exactly what the concept of a function is. Here, the area depends on the radius & is said to be ‘a function of the radius’. The area, hence, is also called the ‘dependant variable‘ while the radius being independent is called the ‘independant variable‘. Now, depending on whatever value of radius (the input) is, there is a corresponding value of the area(the output).

Every single machine in the world, works on the concept of a function. We give the machine an input & it gives us an output. Lets take a washing machine for example, we give it dirty clothes(the input) & it gives us clean clothes(the output). But, what if we leave our cell phone in our pants while giving it to the machine. It will not execute the function. You will get an output for the clean pants BUT NOT for the cellphone even though they are together.

This means that the washing machine is not programmed to take in the cell phone as an input or we say, the cellphone does not fall into the domain of the machine. Hence, the domain of a function consists of all the values that can be given to the machine as the input & the range are the values of all the ossible outputs obtained from the machine.

Mathematically, a function is described as y=f(x) where x is the input or the independant variable & y is the output or the dependant variable as explained above. Hence, all possible values of x form the Domain of the function & the corresponding output values of y form its Range.

Here is a table where functions are classified into 3 main categories based on their nature.

Algebraic Functions:

Modulus Function:

Fractional Part & Greatest Integer Function:

Signum Function:

Trigonometric Functions – Sine & Cosine Functions

Trigonometric Functions – Tangent & Cotangent Functions

Keeping these basic functions & their graphs in mind, we will move on to Graphical Transformations