Welcome to block 3 Calculus.
My name is Rosa Estela
and I am going to present you this video
talking about functions.
The content of this videos is a
little introduction of mathematical functions,
the definition of a function, the domain,
different types of functions,
one variable or multiple variables
and graphical representation in coordinated axes.
Let's define the inverse of a function
and the concept of limit and continuity.
Not the calculus, but the concept.
Why do we study functions?
Functions are fundamental in mathematics for several reasons.
Functions describe the relationship between variables.
Functions provide a structured way
to organize mathematical ideas.
Function serve as a foundational concept in various
branches of mathematics including algebra, calculus,
and statistics. They are integral to understanding topics, such as limits,
derivatives and integrals. And this is
the basis of Calculus.
Functions can be manipulated through
operations like addition, multiplication and composition.
This allows for the creation of new functions and the exploration
of their properties.
Functions
can be represented graphically, which provides
visual insight into their behavior.
Graphs help in understanding concepts
like continuity, limits, and asymptotic behavior.
And more and
I think the most important, studying functions
encourages abstract thinking and reasoning.
And this is important in mathematics.
So introduction, why do we study functions?
For example let's see this.
The study of traffic on these highways
The real study of this traffic
is represented by a function like this
that you can see.
We can approach,
for example, this function of traffic
by this one and
studying functions and series of functions
we can study the traffic.
So this is like an introduction.
Let's start.
What is a function?
A function f from a set X to
a set Y is an assignment of one element
of Y to each element of X.
Then the set X is called the domain of the function.
If the element y in Y is assigned
to x in X by the function f,
one says that f maps x to y
and this is commonly written
y=f(x). In this notation,
x is the variable of the function.
There are different types of functions.
For example, a real function is a function
from a subset of R to R, where
R denotes the set of real numbers.
A simple example,
the square root of x.
The domain of this function are positive numbers or zero.
Not the negative number.
Another function is
a function of n real variables, for example,
this one with three variables.
We can define these
which represents
a sphere.
Another important example of functions
is the parameterized curves
that are presented by a function
from R to R^n, for example
this one from R to R^2
that every real number
goes to (cost,sint)
if we vary the value of t
we have the unit circle.
A trajectory is the path
that an object with mass in motion
follows through space as a function of time.
So see that this trajectory is an example
of parameterized curves.
We can represent
graphically these functions.
Here we have, with Geogebra,
the representation of the function
that you see in green.
So you put the function
and you have the plot automatically. This is in the plane,
but also we can have
the representation in dimension three. Here,
if you write one function of two variables, x and y,
x squared
plus y squared
this is the surface
represented also with Geogebra,
in this case.
See that, if you have different
surfaces, like for example this cone,
and you want to intersect with a plane,
then you have that the intersection
is a plane curve.
You can see here, also working with Geogebra.
You can rotate the figure
and enjoy doing that.
In our example,
the intersection is a circle.
If you need, you can also have the equation with Geogebra.
So what's the inverse of a function?
The inverse of a function f
is another function
that undoes the operation of f.
The inverse exists,
if and only if, f is bijective
and if it exists
is denoted by f`-1
If f is one function
from X to Y
then the inverse function is the function
that for each point y
f^-1(y)
is the unique element x in X such
that f(x)=y
The next property
is satisfied
using the composition of functions, if
we compose f with its inverse, we have the identity
in both senses of the composition.
Graphically
if we have a function f in the plane,
the inverse of this function is
the symmetric function
with respect
to the line y=x.
You can see this
in the image
f in red and its inverse in blue.
An important concept
is the limit of a function,
that we note like that.
What's the limit of the function f
when x approaches p?
This limit is the number L
with the following property.
Given any target distance from L there is a distance
from p within which the values of f(x) remain
within the target distance.
Mathematically you have here the definition.
This is the formal definition of limit
and is the same that we are saying before.
An example,
an asymptote
is a line that the curve approaches
at the edges of the coordinate plane.
Here, for example,
you can see the function one divided by x
and you you can see that
if we approach by
one x
approach, one, the limit is one.
Because you see the point
one, one is a point of the function,
but if we approach by zero
then the function goes to infinity.
So the line x equals zero is an asymptote.
Another important concept of functions
is the continuity. A function f,
in this case, a real function, is continuous
in some point p of its domain
if the limit when we approach to p
is exactly the value of f(p).
Graphically,
you can see in this example,
this given function is always continue
because you can plot the curve without problem
in a continuous manner.
Let's understand another example
when the function is not continuous.
We say that the
function is discontinuous in some point p
if the definition of continuity is not hold.
For example, in this example, function in red, you have this function.
See that happens in zero.
There is a jump.
So, if definition
of continuity
is not fullfiled
there are 3 possible cases of discontinuity
removable discontinuity, jump discontinuity
or essential discontinuity.
In our example is jump discontinuity
as you can see in the graphic.
You can study more about functions,
for example, in these references that we have here. Two books
and the MOOC El Lenguaje de la IngenierĂ­a,
you have videos and exercises with more about functions.
In English,
you have also MathCom T.I.M.E,
you can have more information
and to calculate or to plot functions, you can go to Geogebra,
or WolframAlpha.
So enjoy!
Thank you.