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Hi!

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My name is Vaibhav, and today we will learn
about functions.

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Functions are very important concept, both
for your engineering entrance exams and also

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to understand various concepts in mathematics.

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It's a very fundamental concept, and is used
to describe a lot of phenomenon, including

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physical and chemical phenomena.

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And, also a lot of chapters we will learn
further on such as limits, derivatives, continuity

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or be it any other chapter in mathematics,
knowing functions is very important.

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It is a very fundamental chapter.

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It has various definitions, and so a lot of
use for the next topics that you will learn.

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So, we will start with an intuitive definition
of functions.

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Something that we will try to understand using
examples, and then we will try to go into

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a more mathematically precise definition.

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Functions are used to describe, a special
relation

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between two variables.

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So, here the word "relation" is a mathematical
concept that we will understand later; but

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this (functions) is a very intuitive concept.

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We have two variables of interest, and we
are trying to define what is the relationship

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between them.

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And, that is something which is important
in a lot of places, because you always have

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something such as an input to a system and
an output to a system and you want to figure

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out what the system is doing.

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Functions come in handy at that place a lot.

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This is a very intutitive description of functions.

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In order to define it for our use, we ... mathematicians
... use different types of descriptors for

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functions.

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Typically, functions are described using a
graph, a rule or an equation.

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Now, we will see how this is done.

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When you have two variables either you can
drive a graph, a plot OR you can have a very

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intuitive rule which you can easily describe
OR at times as we will see more and more we

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will be using equations to describe a function.

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So, let's begin with a very simple example.

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We will take life of a typical student who
is studying for an exam and going to school.

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We want to figure out who his or her day goes.

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Let's call the student Sonu, and we want to
find out how Sonu's day went.

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And, the variable that we are interested in
is "How far Sonu is from home?"

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Variable 1 is time.

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Variable 2 will be
Sonu's distance from home.

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Now, let's go to the first definition that
we wrote.

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So, our first description for function is
that function is a "special" relation between

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two variables.

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And, here what we have are two varibles of
interest - one is "time of the day" and (the

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other is) "how far Sonu is from the home."

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So, let's try to make plot for a typical day.

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I will try to remember how my school days
were, and try to make a plot based on that.

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On x-axis I will draw time of day, and on
y-axis I will draw distance from home.

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We will try time count from midnight, and
in 24-hour format.

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A

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day has 24 hours.

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And, then we want to figure out how the day
went.

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In morning, Sonu was home till 6 o'clock.

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Then he when to school, so the distance was
how much farther the school was.

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He was at school till roughly 2pm, which is
14 hours.

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Then let's say he goes to a friends place,
and spends some time there.

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Then he comes back home, and spends the rest
of the time at home.

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So, here we see very intuitively we can describe
the relationship between two variables using

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a plot.

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This plot is essentially describing a function
of where Sonu is throughout the day - how

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far Sonu is from his home - throughout the
day with respect to time of the day.

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Similarlt, we can draw such graphs or such
relations for different variables.

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So, taking an example from physics.

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Let's say we want to study a similar thing
for an object which is moving with a constant

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velocity.

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So, let's take another example.

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And if we want to draw a plot of distance
of this object from the starting point it

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will be something very similar to this.

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The nature of the graph will be different,
but essentially we are plotting similar things.

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Variable one will be time.

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And, variable two will be distance from start.

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And (from physics) we know if an object is
moving with a constant velocity ... it can

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be described by an equation.

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Let's say we denote the variable two as 'd',
and this is described by 't'.

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And let's say that the speed is one which
is given.

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This is an example where we described the
function as an equation.

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So, we saw that a function can be described
by a graph, a rule.

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At first the rule was where you are from home.

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Or it could be a rule like object moving with
a constant speed.

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At times we also use equations.

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And, they are interchangeable.

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We can write this rule as an equation and
this equation as a graph.

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This is how distance varies with time.

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These are some examples of function, which
occur in day to day activities.

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And in these examples we saw that function's
input was time, but thay doesn't have to be

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the case.

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Functions can have different notions of variable
one and variable two depending on the application.

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Another interesting way to look at function
is as an input output relation.

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Here, in the examples that we studied we had
two variables.

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We can see them as a relation between input
which goes through the function and out comes

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the output.

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So, in example 1 input was time of day.

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And, output was distance from home.

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Similarly, in example two the input was time
in hours and output was distance from starting

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location.

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So, we have seen multiple ways of intuitively
describing a function.

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Either it could be a special relation between
two variables, it could be described using

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a graph, rule or equation ...

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It could be thought of as a system which converts
an input to an output.

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And whenever we have anything of this sort,
we can use functions to describe it.

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Once we have these functions, they are pretty
useful and they can be manipulated to figure

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out to find out properties of a system.

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So far, we have seen a few examples.

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Now, I would request you to take some time
to think about other functions that you encounter

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in daily life.

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For those of who you like sports, you can
think of something in cricket.

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Your favorite hits a ball and you want to
figure out how fast the ball is moving.

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So, you can have time after which the ball
was hit and the speed of the ball.

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It does not necessarily have to be speed or
time, it can be something all together different.

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Let's say you are going from Bombay to Delhi,
and the number of passangers change at each

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station.

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Based on the distance from Mumbai, you can
figure out the number of passangers in the

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train.

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In the train example, distance is the input
variable or the variable 1.

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And output variable is number of passengers.

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And similarly you can find other functions
in your day to day activities.

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They occur in various physical and chemical
phenomena.

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So, a lot of things that you are learning
in physics and chemistry - functions are also

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useful there to describe those phenomena.

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Again, take some time to think about examples
of functions.

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Also, one thing that is important to see here
is what makes variable one go on the x axis

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and variable two on y.

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Is it possible to flip the variables on the
graph?

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Why is variable one on the x axis and variable
two on the y axis?

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One reason that you could think of is that
variable one is input and variable two is

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the output; but there are other reasons that
we will see as we go ahead.

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This was a very intuitive definition of function,
now let's study it from a mathemaically precise

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perspective.

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For that we will go back to our orginial statement
- "functions are used to describe special

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relation between two variables.

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"

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Here we have a few mathematical terms, we
have relations and w have variables.

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We will try to see how they relate to the
mathematical concepts that we already know.

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One thing that we have already learnt is sets.

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Sets are a collection of items.

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So, let's say we have sets A and B.

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A and B could be any sets of objects of interest.

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Let's denote elements of A with small a, and
that of set B with small b.

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Once we have these sets, we have variables
from those sets.

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Whenever there is any relationship between
these variables, we can define them using

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certain mathematical constructs.

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One construct is the Cartesian products of
two sets.

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So, a Cartesian products of two sets A, B
is given by A cross B. And, it includes all

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the elements a comma b.

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Let's say A has two variables a1 and a2.

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And, set B has one variable b1.

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Then, A cross B will have two variables (a1,b1)
and (a2,b2)

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(cut near )

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But, if set B had another variable b2.

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Then the total variables will be four.

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Cartesian products of two sets includes number
of elements of set A multiplied by number

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of elements of set B. T

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This is a very big set, and a Relation is
defined as a subset of the Cartesian product.

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So, let's say for some reason you are only
interested in the subset of the Cartesian

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product set.

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We call it by R.

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R let's say has (a1, b1) and (a2, b2).

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Any subset of the cartesian product is a relation.

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Now, that we
have defined the important concepts that we

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need for the precise defintion of fucntions,...
we will see in the next lecture how we combine

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sets, cartesian products and relations - specifically
relations to figure out what is special about

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functions.