Calculus - Functions

Hi everyone! This is the first video in a short series of videos in which I’ll be covering
the basics of calculus
If you follow along with me, you’ll learn how to find the rate of change of a function at
a point
as well as the area between the graph of a function and the x-axis
These are the two fundamental problems to which calculus provides an answer and
everything we learn will be based on these two concepts
Today we’re going to learn about functions, which, at least for now, are really the only
kinds of equations we're interested in
In algebra, we wrote equations as y equals, and then something
involving x
If an equation is a function
we can replace the y with f(x)
When we used to write coordinate points as (x,y), we can now write them as (x,f(x))

A function is a special kind of equation that tells us about the unique relationship
between two variables
x and y
The reason functions are so special and interesting to us
is because a function will give you one
and only one y
for any x you put into it
you put one value in
you get one value out
oftentimes we like to think about a function machine
that has input side and an output side
The machine itself is the equation that relates x and y
When you put an x-value into the input side of the machine, it will calculate
exactly one y-value and spit it out on the output side of the machine
Because of this kind of one-to-one relationship between inputs and outputs
you can think of a function as being made up of its entire set
of input-output pairs
each pair is one point on the graph of the function
The graph, therefore, is a picture of all the input/output pairs, and it can
help us visualize and predict the output that the function will return for any
input
Now, if you’re taking calculus, it probably means that you’ve taken lots of other
math classes to get to this point
which means that somewhere along the line you’ve probably talked about domain and range along the way
The concepts of domain and range are very closely associated with the
concepts of a function
and the one to one
input-output relationship that it models
you can think of the domain of a function as the set of all of the possible input values
depending on the function
some values will be impossible to input
for example
you remember from previous math courses that at least with real numbers you can't
take the square root of a negative number
so if you have a square in your function
you won’t be able to input any value into your function machine
that will make the value underneath the square root sign
negative
these inputs just won't work
you can't put them into your function so they're not in the domain of your
function
The range of a function is completely dependent on the domain
because the range is made up of all of the values you can get out of the
function machine
as a result of inputing all the values in the domain
For example, if your function is x squared, you can input any value you want
without breaking the laws of mathematics or your function machine
so the domain is the set of all real numbers
The range of the function, however, is only positive real numbers
Regardless of whether you input negative or positive values into the function, because
you’re squaring each value, you can only get positive numbers out of it, and so the
range of the function is only positive real numbers
Functions are NOT any equation that can return multiple output values for the same input value
circles or a great example of non-functions
If we pick any x-value inside the circle
the equation of the circle returns two output values
for the single input value
remember
by definition, an equation is only a function if it returns exactly one
output
for every input value
so we can clearly see that a circle
will never be a function
this brings us to a great way to test for functions
it's called the vertical line test
and it says that, if you draw a perfectly vertical line anywhere in the domain of a
function
it will cross the graph only once
If you can draw a perfectly vertical line
that crosses the graph more than once
then the points where the line intersects the graph
represent the multiple output values for the single input value, and the graph therefore
cannot represent a function
Keep in mind also that a function can itself be comprised of multiple
functions
This polynomial function is a great example
the entire thing represents a function
but so do each of the four terms inside it
is taking separately
each one would pass the vertical line test
this function is therefore what we call a combination
of other functions
functions can also be compositions of functions
where one function is nested inside another
For example, e to the x is a function, and x squared is a function
e to the x squared
is a composition of functions where x squared
is nested inside e to the x
Sine of x squared + 1
is another good example of a composition of functions
Next time we’re going dive into the real foundation of calculus
and talk about limits and continuity
I'll see you then
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