Look at this bird flying.
Can you think of some functions related to its motion ?
We saw two such functions earlier.
The distance travelled as a function of time
and its speed too as a function of time.
But we also know that as the altitude rises,
the atmospheric pressure decreases.
So, the atmospheric pressure felt by the bird
is the function of its height above the ground.
We previously saw that when we have two quantities,
‘X’ and ‘Y’, such that for each value of ‘X’,
there is one value of ‘Y’…
then we say that ‘Y’ is a function of ‘X’.
So here ‘X’ could be time and “Y’ could be the distance.
‘X’ can be the altitude and ‘Y’ the atmospheric pressure.
‘X’ could be time, and ‘Y’ the speed.
Can you think of other functions like these?
Let me give you one example!
What do you think the POSITION of a book in a library
is dependent on ?
Each book in a library is labelled using letters
and numbers like these.
And these letters and numbers
could be based on the subject,
or the genre or edition maybe!
So the position of the book
is dependent on some letters and numbers.
Any more examples you can think of ?
Share your thoughts in the comments section below.
But just knowing that two things
depend on each other isn’t very useful to us.
We need to know HOW they are related to each other.
Previously we saw that this relationship
can be given by equations like these.
This equation says that the value of the
dependent variable is the square of the value
of the independent variable.
But can a function like the position of a book
in a library be represented like this ?
No, we cannot do that, right ?
We cannot represent this function by an equation
like this.
In calculus we only deal with functions
between two things that can be QUANTIFIED.
Such quantifiable functions
can also be represented visually.
Do you remember the question we asked
in the previous video ?
We asked what is common between
this equation and a flashlight or a dish satellite.
Did you find anything ?
The mirror in the flashlight and this dish are
similar curved surfaces called parabolic surfaces.
The special thing about such surfaces
is that in the case of a satellite,
all the electric signals falling on the dish
get reflected to this point called the focus.
And in the case of flashlight, it’s the opposite.
The bulb inside is located at the focus point.
When we turn it on,
all the light rays instead of spreading in all directions,
get reflected in the forward direction.
Now look at the cross section of the curved surface.
Notice that the whole surface is made up of curves
which look like this.
We have seen earlier
that this type of a curve is called a parabola.
And here’s the connection.
This curve is the geometrical representation
of this equation.
Or we can say that this equation
is the algebraic representation of this curve.
Let’ s say we draw ‘X’ and ‘Y’ axes like this
The values of independent variable ‘X’
are represented by the points on the ‘X’ axis.
And the values of the dependent variable ‘Y’
are represented by the points on the ‘Y’ axis.
Here, the ‘X’ and ‘Y’ co-ordinates of the points
on this curve are related basis this relation.
If we take different values of the constant ‘K’ here,
we will get different functions.
And for each function,
we can draw a curve corresponding to it.
Just by looking at them we can figure out
how the two functions behave relatively.
As the value of the variable ‘X’ increases,
the value of the variable ‘Y’ increases relatively faster
in this case.
Visualizing a function
gives us an INTUITION about how it works.
Now let me ask you a question.
In our previous video,
we found out the rate of change in the
dependent variable the dependent variable with
respect to the independent variable for this function.
We get this new function for the rate.
So here,
can you tell me what this rate means geometrically ?
Think about it and share your thoughts
in the comments section below.
So in this lesson, we saw that a function can be
represented by its equation and its graph.
Now look at this curve.
Can you tell me
whether this curve represents a function or not?
We will continue this in the next part.
This curve does not represent a function.
Let us understand why ?
For now,
let’s first assume that this curve represents a function.
We usually denote a function in this way.
It is read as ‘F of X’.
This notation is made up of three things.
First , ‘X’ here represents the independent variable.
Second , ‘F’ represents the function itself.
Third , ‘F of X’ together represents the
dependent variable which we earlier denoted by ‘Y’.
Another way to write it is ‘Y equal to F of X’.
So this tells us that the variable ‘Y’
is some function of the variable ‘X’.
And that function is denoted by the letter ‘F’.
Such type of notations also help us
when we want to talk about two different functions
between the same variables .
We can denote them using different letters like this.
Now let’s come back to our question.
Look at this point on the ‘X’ axis.
We denote it by ‘X one’ .
Now we see that related to ‘X one’
there are two values of the variable ‘Y’.
And that’s why this curve does not represent a function.
We have seen that a function is a relation between
two variables, such that for each value
of the independent  variable there is only
ONE value of the dependent variable.
But if we draw a line parallel to the ‘Y-axis’ here,
we see that for one value of ‘X’,
there are two values of ‘Y’.
So this is not a function.
Drawing a vertical line like this helps us.
For example look at this curve.
If we traverse this curve through a vertical line,
we see that at some points,
the line intersects the curve at more than one point.
So this curve cannot represent a function.
Checking whether a curve represents a function in
this way is called the vertical line test.
Now let’s get back to this function
and its geometrical representation we saw earlier.
Notice that in this equation, for any value
of the variable ‘X’, we get one value of variable ‘Y’.
This can be easily seen from its
geometrical representation.
But now look at this function.
What will be the value of the
‘Y’ variable for ‘X equal to one’ ?
For ‘X equal to one’ here, we get ‘one divided by zero’.
Anything divided by zero is undefined!
Now if we draw its graph, it will look like this.
So we see that this function
is not defined for ‘X equal to one’.
But for for all other values of ‘X’,
we will get one value of ‘Y’.
So except for ‘X equal to one’
the function is defined for all other values of ‘X’.
Therefore we always have to mention the collection of
all the acceptable values of ‘X’.
It is called the DOMAIN of the function.
And the collection of all the values of ‘Y’
related to the values of ‘X’ in the domain
is called the RANGE of the function.
Now we have an idea about the concept of functions!
Let’s say we have sets ‘X’ and ‘Y’ both of numbers.
And we are given this relation between the two sets.
We can see that related to each number in the ‘X’ set
there is only one number in the ‘Y’ set.
So this relation is a function.
It can be written like this .
The domain of the function is the numbers in the set ‘X’.
And the range of the function
is the numbers in the set ‘Y’.
But look at this relation between two sets of numbers.
Is this a function ?
If it is , then what is its domain and range ?
Yes it is a function.
Related to each number in set 'A'
there is only one number in set 'B'.
It doesn't matter if there are more numbers
in set A that are related to a number in set B.
What about the domain and the range ?
Here the domain is this set of numbers.
But now notice that this number in set ‘B'
is not related to any number in set ‘A'.
So it is not in the range of the function.
The range will be the set of these 2 numbers.
So we can now talk about functions in a general sense.
Let’s move on to a new page!
Here is a set of things.
And related to each of its elements,
there is ONLY one element in some other set.
This relationship is called a function.
And how they are related is usually given by an equation
or shown pictorially with the graph of the function.
Now let me ask you a question.
Why exactly are we looking at the concept of functions
in Calculus ?
As we move along, we will realise that all the problems
we discussed about in our previous videos, can be
framed in terms of functions between two variables.
Let’s say we are given this graph of a function.
Then finding the solutions of the problems
we discussed earlier, boils down to two things.
First , to find out the steepness of a tangent line
at a point on the curve.
Differentiation is the process that is used here.
Now the Now the second thing is to the find the
area of a particular region like this under the curve.
Integration is the process used here.
In the upcoming lessons
we will see how exactly this is done.
To stay updated,
subscribe to our channel by clicking here !
Happy learning !
