Overview on Basic Mathematical Functions

4 min readJun 29, 2021

1. Function

1.1 What is Function

Before going into details about “Function”, let’s talk about a simple machine
first. If it is asked, “What does a simple machine do?” the very first thing
that should come to your mind is, “It takes an input, process it, and gives an
output”. If it is represented through a diagram, then it should look like,

So, now let’s get back to the topic “Function”. “What does a Function
do?” The answer is, “A Function exactly works like a Simple Machine”.
You can get a clear idea when you see the below diagram.

1.2 Domain & Range

In very simple words, “All the possible inputs for a function” is called the domain of that particular function. A domain represents all the values of input or x.
On the other hand, “All possible outputs due to given input” is called
the range for a particular function.
Now, it can be asked, “What is the relationship between Domain and Range?”
Well, the answer is, Range is the corresponding output for a particular Domain/Input which is processed by the given function f(x). The given diagrams will make the idea clearer.

Figure 3: Relationship between Domain & Range

1.2.1 Interval Notations of Domain & Range

Here is a brief overview of the interval notations through a table (learn more click here).

1.3 Graphing of Functions

It is familiar to almost everyone “How to Graph a Function?” However, I
will give a short overview of it. You can see it in the below figure.

Figure 5: Graphing of a Function (step by step)

1.3.1 Squeezing & Stretching
When a graph Squeezes: Whenever a function is being multiplied by
a whole number it squeezes. With the increment of the whole number, the ‘Graph Squeezes More’ gradually.

When a graph Stretches: Whenever, a function is being multiplied by the fractional number it stretches. With the increment of the fractional
number, the ‘Graph Stretches More’ gradually.

1.3.2 Shifting

Vertical Shifting: When an additional number (it can be whole or fractional) is being added to the function the graph shifts upward.

Horizontal Shifting: When an additional number (it can be whole or
fractional) is being subtracted from the function the graph shifts downward.

1.3.3 Change of the Origin

By adding two units to the x-axis, it shifts the curve to the right by two
units. On the other hand, by subtracting two units to the x-axis, it shifts
the curve to the left by two units.

1.4 Function Composition

Function Composition is a convenient way for building up complex functions by combining simple functions. It is done in a shape of a chain
design. Intuitively, the output of the previous function is fed as input of the
current function and this process continues up to n number of times or the
number of times it is needed.

Figure 11: Simulation of Function Composition

Now let’s see a complex example.

Problem:

Solution:

1.5 Inverse Function

The inverse of a function tells you how to get back to the original value. We
do this a lot in everyday life, without really thinking about it. For example,
you are a new student at your university. You are looking for one of your
batch mates but you do not know any basic contact information (Name,
Contact Number) of that person. The only thing you know about him is
“He is enrolled in the same section as yours in the mathematics course”. So if you think mathematically,