8
Aug

4. Decomposition, Abstraction, and Functions


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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right
everyone let’s get started. All right good afternoon on
this rainy, rainy sad afternoon. So– I’m glad we’re
inside though– all right so Lecture 4 of 6.0001 in 600. Quick, quick recap of
what we did last time. So last time we did a little
bit more string manipulations, and then we saw how you can
use for loops over strings directly. So instead of having for
loops that iterate over range– so 0, 1, 2,
3, 4, and so on– you saw that it was more
powerful to sometimes use for loops that iterate over
string objects directly. So that was the first
half of the lecture. In the second half, we started
looking at different ways that you can implement the
different implementations to the same problem. So we saw the problem of
finding the cube root, and we saw some implementations. We saw the Guess
and Check method, and the approximation method. And then we looked
at what I thought was the most powerful method,
which was the bisection method. And this one, if you
remember, I played a game with someone in the
audience where I guessed a number between 0 and 100. And we saw that I was able
to guess that number really, really quickly using
the bisection method. And that’s the
method that you’re going to implement– that you
are currently implementing– in your problem set. OK so today– so that sort
of finishes introduction to some of the more basic
mechanisms in Python. And today we’re going
to talk about how to structure your programs
such that you write nice, coherent code– reusable
code– by hiding away some of the details in your code. And to do that we’re going
to look at these things called functions. All right so just
stepping back and sort of getting a high-level view of
how we write the code so far. So so far the way that
you’ve been writing code for your programs
is you open a file, you type some code to solve
a particular problem given, like in your problem
sets, each file contains some piece of code, you
have sequences of instructions that contain maybe assignments,
loops, conditionals, and so on and so on. But really you have one
file that contains each code and you write everything
in that particular file. But this is OK for
smaller problems that we’ve been seeing
so far, but when you’re starting to write
large pieces of code it’s going to get really
messy, really quickly. So think about if you
want to use a for loop in one part of
your code, and you find it useful to use
that same for loop in another part of your code. Some point in the future as
you’re debugging your code, you might want to change
your original for loop, you have to figure out
all the other places where you’ve used that type
of for loop for example. So as you’re scaling
your code, you’ll find it harder to keep
track of these details. So this is where functions
will come into play in today’s lecture– will help you out. So if you want to be
considered a good programmer, a good programming style would
be to not necessarily add lots and lots of lines
of code, but really to add more functionality
to your programs. So how many different things–
how many different features– can your program do, rather
than how long can your code be. And that’ll help you
later on look at your code if you need it for
a future class, and it’ll help others if they
want to look at your code later on if they find it useful. So today we’re introducing
this idea of functions. And functions are
mechanisms to achieve decomposition and abstraction. So these are two
key words here that are going to pop up
in today’s lecture and also in future lectures. So before I introduce
decomposition and abstraction in the context of
functions, let’s first take a look at just
sort of a real-life example. So let’s take a projector. I’m using one right now. Quick show of hands. If I give you all of the
electronic components that are part of a projector–
resistors, a fan, a light bulb, a lens, the casing, all of
the different parts in it. Who here would be able
to build a projector? Do I see a hand? No? Ooh oh yeah nice! You can also lie. I won’t know the difference. But if you can do that,
I’d be very impressed. All right so you can’t really
put together a projector right? Another show of hands. If I gave you a projector
that’s fully assembled and I gave you a
computer, for example, who would be able to maybe
figure out within let’s say an hour how to make
them work together? Good, a fair bit of the class. That’s perfect. That’s exactly the answers
I was trying to get at here. So none of us really
know how a projector works– the internals–
but a lot more of us know how to work a
projector, just given maybe a set of basic instructions
or just intuitively speaking. So you see the projector
as sort of a black box. You don’t need to know how
it works in order to use it. You know maybe what inputs
it might take, what’s it supposed to do at a high level. Take whatever’s on my screen and
put it up on the large screen there, just magnify
it, but you don’t know how it does it– how
the components work together. So that’s the idea
of abstraction. You don’t need to know how
the projector works in order to use it. OK that’s abstraction. The other half of that
was decomposition. So let’s say that now,
given a projector, I want to project a very,
very large image down on a very large stage. For example, this is
from one of the Olympics. It’s a stage of what,
like 10 football fields, something like that? Something massive. You could build one
projector that’s able to project a
very large image, but that would be
really expensive and you’d have to build
this one projector that’s used for this one time. So instead what
you could do is you can take a bunch of
smaller projectors and feed different inputs
to each one of them. And as you’re feeding
different inputs, each one’s going to
show a different output. And then you’re going
to be able to have all of these different
projectors working together to solve this larger
problem of projecting this really cool image
on a very large stage. So that’s the idea
of decomposition. You take the same projector,
feed it different inputs, it does the exact same
thing behind the scenes, but it will produce
a different output for each one of these
different inputs. So these different devices
are going to work together to achieve the same
common goal, and that’s the idea of decomposition. So these is where I apply
to the problem of projecting large image, or a
projector in general, but we can apply these exact
same concepts to programming. So decomposition is
really just the problem of creating structure
in your code. In the projector example,
we have separate devices working together. In programming, to
achieve decomposition you’re dividing your code
into smaller modules. These are going to
be self-contained, and you can think of them as
sort of little mini-programs. You feed in some input to
them, they do a little task, and then they give
you something back. They go off and do
their thing and then they give back a result. These modules can be used
to break up your code, and the important thing
is that they’re reusable. So you write a module once–
a little piece of code that does something
once– you debug it once, and then you can reuse it
many, many times in your code with different inputs. Benefit of this is it
keeps your code organized and it keeps your code coherent. So functions are going to be
used to achieve decomposition and to create
structure in our code. We’re going to see functions
today in this lecture, and in a few weeks, you’re
going to actually see– when we talk about object
oriented programming– how you can achieve
decomposition with classes. And with classes you can
create your own object types like adding some floats. You can create your own object
types for whatever you want, but that’s later. OK so decomposition is creating
structure in your code. And abstraction is the idea
of suppressing details. So in the projector example,
remember, abstraction was you didn’t need to know
exactly how the projector worked in order to use it. And it’s going to be the
same idea in programming. So once you write a piece of
code that does a little task, you don’t need to rewrite
that piece of code many times. You’ve written it
once, and you write this thing called a function
specification for it, or a docstring. And this is a piece of
text that tells anyone else who would want to use it
in the future– other people, maybe yourself– it tells
them how to use this function. What inputs does it take? What’s the type of the inputs? What is the function
supposed to do? And what is the output that
you’re going to get out of it? So they don’t need to know
exactly how you implemented the function. They just need to know
inputs, what it does, what’s the output. Those three things. OK so these functions are
then reusable chunks of code. And we’ll see in a few
examples in today’s lecture how to write some and
how to call functions. And as we’re going
through today’s code, I want you to sort of
think about functions with two different hats on. The first hat is from someone
who’s writing the function. So in the projector
example, someone had to build the
first projector. Someone had to know how to put
all these components together. So that’s going to be
you writing a function, so you need to know how
to make the function work. And then the other hat
is you as someone– as a programmer– who is
just using the function. You’re assuming it’s already
been implemented correctly, and now you’re just
using it to do something. So these are some of the
function characteristics and we’ll see an example
on the next slide. So a function’s
going to have a name. You have to call it something. It’s going to have
some parameters. These are the inputs
to the function. You can have 0 inputs or
as many as you’d like. Function should
have a docstring. This is how you
achieve abstraction. So it’s optional, but
highly recommended, and this is how you
tell other people how to use your function. Function has a body, which
is the meat and potatoes of the function– what it does. And a function’s going
to return something. It computes its thing and
then it gives back– spits back some answer. OK here’s an example of
a function definition and a function call. Function definition is up here. I’ll just draw it here. This is the function
definition up here. And this is the
function call down here. So remember, someone has
to write the function that does something to begin with. So this is how you
write the function. The first is
whoops– the first is going to be this def keyword. And def stands for–
it tells Python I’m going to define a function. Next is the name
of the function. In this case, I’m calling
the function is_even. And the function
name should really be something descriptive. Whereas someone who is
just using this function or looking at it can
pretty much tell what it’s supposed to do without
going a lot farther than that. They’re just
looking at the name. And then in parentheses you give
it any parameters, also known as arguments. And these parameters are
the inputs to the function. And then you do colon. OK so this is the first line
of the function definition. And after this,
everything that’s going to be part of the function
is going to be indented. The next part is going
to be the docstring, or the specification,
and this is how we achieve abstraction
using functions. Specification, or the docstring,
starts with triple quotes and ends with triple
quotes, and you can sort of think about this as
a multi-line comment. It’s just going
to be text that’s going to be visible to
whoever uses the function, and it should tell them
the following things: What are the inputs to the function? What is the function
supposed to do generally? And what is the function
going to give back to whoever called it? The next part is going to
be the body of the function. We’ll talk about what’s
inside it in the next slide. And that’s it. That’s all for the
function definition. def blah, blah, blah, indented,
everything inside the function. So this is you writing
the function definition. Once the function
definition’s written, you can call the function. And that’s this part down here. And here, when
you call function, you just say its name, and
then you give it parameters. And you give it
as many parameters as the function is expecting–
in this case, only one parameter. So what’s inside
the function body? You can put anything
inside the function body. You remember,
think of a function as sort of a small procedure
or a little mini-program that does something. So you can do anything
inside the function that you can do in the regular
program– print things, do mathematical
operations, and so on. The last line is the most
important part of the function though. And it’s this return statement–
that’s what we call it. So it’s a line of code
that starts with return, which is a keyword. And then it’s going
to be some value. Notice this is an
expression here– i%2==0 is an expression
that’s going to evaluate to some value. And as long as this
part is something that evaluates some value,
it can be anything you want. And this line here return
something tells Python, OK after you have finished
executing everything inside the function, what
value should I return? And whoever called
the function is going to get back that
value, and the function call itself will be
replaced by that value. OK so let’s look at an example. I’m going to introduce
the idea of scope now. And scope just means– is
another word for environment. So if I told you that you
could think of functions as little mini-programs,
the scope of a function is going to be a completely
separate environment than the environment
of the main program. So as soon as you
make a function call, behind the scenes
what Python says is, OK I’m in the main program
but I see a function call. I’m going to step out
of this main program. I’m going to go off into
this new environment. I’m going to create entirely
new set of variables that just exist within this environment. I’m going to do
some computations. When I see the return, I’m going
to take this one return value. I’m going to exit
that environment, and then I’m going to come
back to the main program. So as you’re entering
from one scope to another, you’re sort of passing
these values back and forth. So when you’re entering a scope,
you’re passing a variable back into the function. And when the
function’s finished, you’re passing a value
back to whoever called it. So once again, this top part
is the function definition. And any arguments for
the function definition are called formal parameters. And they’re called
formal parameters because notice they don’t
actually have a value yet. In the function
definition, you’re sort of writing the function
assuming that, in this case, x is going to have some value. But you don’t know
what it is yet. You only know what
value x takes when you make a function call down here. So this is your
function definition, and then later on in
your main program, you might define some
variable x is equal to 3. And then you make
a function call. f of x here is
your function call. And it says, OK I’m
calling f with the value 3, because x takes the value 3,
and then I’m going to map 3 into the function. The values that are passed
into the function call are called actual parameters,
because they’re going actually have a value. So let’s step through this
program– this small program– and see what exactly happens
behind the scenes in the scope. And if you’re just
starting to program, I think it would be
highly valuable if you take a piece of paper as you’re
doing some of these exercises and you write down
something similar to what I’m going to go through here. I think it’ll help
a lot, and you’ll be able to see exactly
step-by-step what variables take what values and
which scope you’re in. So here we go. When the program
first starts, we’re creating this global scope. It’s the main program scope. In the main program
scope, the first thing that Python is going
to see is this part here– def f of x and
then some stuff inside. This tells Python I
have a function named x, but I don’t care what’s
inside the code yet. I don’t care what’s inside
the function definition yet, because I haven’t
called the function yet. So to Python it’s
just some code just sitting in the global scope. So whenever you see
def, you’re just putting some code in there. Then you go onto the next
line– x is equal to 3. So in the global scope, you now
have also a variable x is 3. And then the next
line– z is equal to f of x is a function call. As soon as you hit
a function call, you create a new scope–
a new environment. So we’re temporarily leaving
the global scope and sort of portaling into a
new scope, where we’re going to try to figure out what
this function’s going to do and what it’s going to return. So the first thing you do
is you map the parameters. So x here– I’m
calling f of x with 3– so first thing I’m
doing is I’m mapping every one of the parameters in
the definition to their values. So first thing I’m doing
is x gets the value 3. Next line here is x
is equal to x plus 1. So we’re still inside
the function call f, so x gets the value 4. We’re printing this and
then we’re returning x. So in the scope of
f, x is equal to 4, so we’re returning
that value back to whoever called it,
which was this function call within the global scope. So this part right here– f
of x, which was the function call– gets replaced with 4. So inside the main
program, z is equal to 4. And that’s how we pass
parameters into the function, and we got a parameter
back from the function. As soon as the function
returns something, the scope that you were in
for the function gets erased. You forget about every variable
that was created in there, delete that scope, and
you’re back to wherever you started calling it. One warning though. So what happens if there’s
no return statement? I said that every function
has to return something. If you don’t explicitly
put a return statement, Python is going to
add one for you. You don’t have to do this. And it’s going to actually
have return None– N-o-n-e. And None is the
special type– None is the value for a special
type called NoneType, and it represents the
absence of a value. What’s that? Not a string. Not a– None is not a string. None is not a string, exactly. It’s a special type. OK so before we go on, I wanted
to go through a small exercise in Spyder just to show
you the difference that None and printing
and returning makes. So here are two
functions that I wrote. One is is_even_with_return. That’s its name, so
pretty descriptive. It’s pretty much the same
code we saw in the slides. It just has this extra
little print thing. It gets the remainder
when i is divided by 2. And it returns whether the
remainder is equal to 0. So it’ll either return a
true or a false– a Boolean. OK so my function call is this:
I’m saying is_even_with_return with a value 3. When I make this
function call, this 3 gets mapped into here–
this variable here– so i is equal to 3. I’m going to print with
return, and then I’m going to say remainder
is equal to 3 percent 2, which comes out to value 1,
because there’s a remainder 1. And I’m going to
return whether 1 is equal to 0, which is false. So this line here
returns false, but am I doing anything with the false? Not really. It’s just sort of
sitting in the code here. So this gets evaluated to false. I’m not printing it. I’m not doing any
operations with it. It’s just sitting there. So it won’t show up anywhere. If I want the result
to show up somewhere, then I have to print it. So that’s what this
next line is doing. So that one should
be straightforward. is_even_without_return’s
a little bit trickier, but not too bad. I have print,
without_return inside here, and then I’m going
to get a remainder is equal to i percent 2. And notice that I’m not–
I don’t have any return. So implicitly, Python’s going
to add a return None for me, like that. You don’t have to add it. So when I make the
function call here, it’s going to do the same thing,
except that return in this case is not going to be a Boolean. It’s going to be
this special None. So this is going to
get evaluated to None. Again I’m not printing it out. It’s just sitting there. If I were to print out
the result of that, you’d be printing out this
value None, which if I run it, you’ll see here it just
prints it out right there. So as you’re doing your next
p set, it’s about functions and you’re seeing these Nones
popping out in some places. Check to make sure that you’ve
actually returned something, as opposed to just printed
something inside the function like we did here. All right so that’s
the difference. And the last thing I want to
mention about this is_even function is how
useful it can be. So notice this is the
function as in the slides, and once you write the
function once, you can use it many, many times in your code. So here I’m using
the function is_even to print the numbers
between 0 and 19, including and whether the
number is even or odd. So notice this
piece of code here, once I’ve written
this function is_even, looks really, really nice right? I have for all the numbers in
this range if the number i is even, this is going to
return a true or false for all the numbers
0, 1, 2, 3, 4. If it’s true, then I’m
going to print out even, and otherwise I’m
going to print out odd. So if I run this,
it’s going to do this. 0 even, 1 odd, 2
even, and so on. So notice using functions makes
my code really nice looking. If I wasn’t using
functions, I’d have to put these two lines
somewhere inside here and it would look a
little bit messier. So I’ve said this
maybe once or twice before: in Python
everything is an object. Might not have meant
anything back then, but I think you’re
going to see what I mean using this particular example. So if in Python everything’s an
object– integers are objects, floats are objects, even
functions are objects. So as you can pass objects
as parameters back and forth as function parameters, you
can also pass other functions as parameters. Let’s see what this means. So we have three function
definitions here– func_a, func_b, and func_c. And then I have three lines of
code here in my main program. So I have one called a
func_a, one called a func_b, and one call to func_c. Let’s trace through, just
like in the previous example, and see what exactly happens. First thing I create
is my global scope. And I have three
function definitions. Again I don’t care
what’s in the code yet, because I haven’t
called the functions yet. Python just knows there’s these
functions with these names that contain some code. After these definitions, I
come to this line here– print func_a. As soon as I make
a function call, I’m going to create
a new scope and I’m going to hop into there. Inside func_a, I’m going to go
and look at what func_a does. It doesn’t take
in the parameters, it just prints out
this message here. And then it leaves; it’s done. There’s no return,
so we return None. So func_a returns
None to whoever called it, which
was that line there, so that is going to be None. Next line. This one right here– print
5 plus some function call. Again I’m going to hop
into func_b’s scope and see what to do there. So first I’m going
to map my parameters. So 2– whoops– 2
gets mapped to y. So inside func_b’s scope, y
is going to get the value 2. That’s the very first
thing I’m doing– mapping all the parameters. Then I’m going to
print this thing here, and then I’m going to return y. So inside func_b,
y has the value 2, and I’m returning 2 back
to whoever called me. So this is the value
2 and I’m going to print 5 plus 2, which is 7. Last one. This is the trickiest. Oop, that popped up. If you think you’ve got
it, try that exercise. But otherwise follow along. print func_c func_a. So I see that I am going
to enter func_c’s scope. So I’m going to look
at what func_c does. First thing I do is I’m
mapping all the parameters. Don’t even worry about
the fact that this is a function right now. Just pretend it’s
x or something. So you say func_a
is going to get mapped to the variable
z inside func_c. So z is func_c. Just mapping parameters
from actual to formal. Then what do we
do inside func_c? We print out inside func_c,
and then we return z. This is the cool part. Inside func_c, z is func_a. So if you replace z
with func_a, this here becomes return func_a
open close parentheses. Look familiar? We did that function
call right there right? So that’s just
another function call. So with that being
another function call, you’re going to
create another scope, and you’re going to
pop into that one. So we’re one, two, I
guess two scopes deep, and we’re trying to figure
out where we’re going. So func_a’s scope is
going to be up here. So what does func_a do? It just prints out this,
and it returns None. So we’re going to
return None to whoever called us, which was func_c. So this line here
becomes return None. And so this line here
is going to return None to whoever called it, which
was this line down here. Oops, I didn’t mean
to cross that out. So that line here is
going to print None. So if you just go
step-by-step, it shouldn’t be too
bad to try to map what happens with variable
names and formal parameters and actual parameters. That’s why I highly recommend
pieces of paper and pens. One last thing I want
to mention about scope before we do another example. So there are three
sort of situations you might find yourself in. The first one is probably
the most typical, and this is when you
define a function. And it’s using a
variable named x in this case that’s also
defined outside of the function. And that doesn’t matter
because of the idea of scopes. So inside the global scope,
you can have variables x. When you’re inside
a different scope, you can have whatever
variable names you want. And when you’re
inside that scope, Python’s going to use
those variable names, so they don’t interfere
with each other at all. So in this example, I’ve defined
a variable x is equal to 1, and then I incremented, and that
doesn’t interfere with the fact that we have a
variable x outside. This one’s a little
bit trickier. I define this function
g, and all g does is access a variable x. But notice inside g, I’ve
never actually declared or initialized a variable x. In this f, I said
x is equal to 1. But in here, I’m
just sort of using x. So this does not
give you an error. In fact it’s OK for you
to do this in Python. Python says, OK
I’m in this scope, but I don’t have
a variable named x, so let me just go into the
scope of whoever called me. So I’m going to just
temporarily hop out of the scope and see is there
variable x outside of me? And it’ll find this
variable x here, and it’s going to
print out its values. So that’s OK. This last example here
is actually not allowed in Python– similar to
this one– except that I’m trying to increment
a value of x, but then I’m also
trying to reassign it to the same value of x. The problem with that is I
never actually initialized x inside h. So if I said– if inside
h, I said x is equal to 1, and then I did x
plus equals to 1, then it would be this
example here– f of y. But I didn’t do that. I just tried to access
x and then incremented and then tried to reassign it. And that’s actually
not allowed in Python. There is a way around it
using global variables. But it’s actually frowned
upon to use global variables, though global variables
are part of the readings for this lecture. And the reason why
it’s not a great idea to use global variables is
because global variables sort of give you this
loophole around scopes, so it allows you to write code
that can become very messy. So using global variables,
you can be inside a function and then modify a
variable that’s defined outside of your function. And that sort of defeats
the purpose of functions and using them in writing
these coherent modules that are separate. That said, it might sometimes be
useful to use global variables, as you’ll see in a
couple lectures from now. OK cool. So let’s go on to the
last scope example. OK this slide is here,
and notice I’ve bolded, underlined, and italicized
the Python Tutor, because I find it
extremely helpful. So the Python Tutor–
as I’ve mentioned in one of the assignments–
it was actually developed by a grad student here,
or post-grad student slash post-doc here. And it allows you to go
through Python, paste a code, go through it step-by-step. Like with each
iteration, it’ll show you exactly what values
each variable has, what scope you’re in,
when scopes get created, when scopes get destroyed,
variables within each scope. So pretty much
every single detail you need to sort of
understand functions. As we’re starting to– you can
see we’ve had couple questions, and these were great questions. So if you’re still trying to
understand what’s going on, I would highly suggest
you take a piece of code and just run it in
the Python Tutor and you should be able to
see exactly what happens, in sort of a similar way
that I’ve drawn my diagrams. In all of the codes for
this particular lecture, I’ve put links to
the Python Tutor for each one of those exercises. So you can just copy
and paste those, and it’ll automatically
populate it with that particular
example, so you just have to click, step, step, step. OK so having made my plug for
Python Tutor, let’s go on. OK so here’s an example. It’s going to show
couple things. One is print versus return,
and also this idea of you can nest functions. So just like you could
have nested loops, nested conditionals– you
can also nest functions within functions. So let’s draw some diagrams
just like before of the scopes. First thing we’re going to
do is when we have a program, we’re going to create
the global scope and we’re going to add
every variable that we have. And then when we
reach a function call, we’re going to do
something about that. So the first thing
in the global scope is this function definition. Again in my global scope,
I just have g as some code because I have
not called it yet. I only go inside a function
when I make a function call. So g contains some code. So we’re done with
75% of that code. Next line is x is equal to 3. So I’m making x be a variable
inside my global scope with value 3. And then I have this
z is equal to g of x. This is a function call. When I see a function call, I’m
going to create a new scope. So here is the scope of g. With the scope of g, I’m mapping
variables to actual parameters to formal parameters. So the first thing
I’m doing is I’m saying inside g what is the
value of actual parameter x? And x is going to be the
value 3, because I’ve called g of x with x is equal to 3. Next, what I see
inside this function– so this is the inside of the
function– is this bit here. It’s another
function definition. Again since I’m just
defining the function and I’m not calling it, all
Python sees is h is some code. I haven’t called
the function h yet, because I’m just defining
it here with def. So that finishes this part here. The next line is x
is equal to x plus 1. So inside the scope of g,
I’m incrementing x to be 4. Then I’m printing out this line. And then I’ve reached here– h. This is actually a function
call, and I’m calling h. As soon as I make
a function call, I’m creating another scope. So I’m temporarily going
out of the scope of g and going into the scope of h. So Python knows that
h contains some code, and now I can go inside h
and do whatever I need to do. So the first– so h doesn’t
have any parameters, so I don’t need to populate
anything like that in there. h does define a variable called
x, which is abc; it’s a string. And then that’s all h does. What does it return? None. I heard murmuring,
but I think None was what you guys were saying. So since there’s no
return statement, h is going to return None. So h returns None. Back to whoever called it,
which was this code inside g. So that gets
replaced with None– the thing that I’ve–
this circled red h here. As soon as h
returns, we’re going to get rid of that scope–
all the variables created within it– and
we’re done with h. So now we’re back into g. And we just finished
executing this and this got replaced with None. We’re not printing it out, so
this doesn’t show up anywhere; it’s just there. So we’re finished
with that line. And the next line is return x. So x inside g is 4,
so 4 gets returned back to whoever called it, which
was in the global scope here. So this gets replaced with 4. So once we’ve returned x,
we’ve completely exited out of the scope of g,
and we’ve come back to whoever called us,
which was global scope and we’ve replaced
z is equal to g of x and that completely
got replaced with 4– the returned value. So that’s sort of
showing nested functions. All right just circling back
to decomposition-abstraction. This is the last slide. You can see if you look
at the code associated with today’s lecture, there
are some other examples where you can see
just how powerful it is to use functions. And you can write really
clean and simple code if you define your own functions
and then just use them later. And the beauty of defining
your own functions that you can use
multiple times later is you only have to debug
the function once right? I know debugging is not
your favorite thing, but you only have to
debug this one thing once, and then you can know that
it’s right and it works well, and you can just use
it multiple times. All right thanks everyone.

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