Defining functions

Functions are a way to wrap up a bunch of code into one word. They make it possible to reuse the same code over and over again, without having to rewrite it. Pretty useful, especially for operations you want to do often. Functions also make it easier to think about your code, because once you know your function works, you never again have to worry about how it works. This enables you to break up problems into smaller, easier problems.

Syntax for defining functions

So how do we define a new function? Use the def keyword, short for "define". Then we'll have to specify a few things:

Let's see an example:

def fifth_fibonacci():
    fib_0 = 0
    fib_1 = 1
    fib_2 = fib_1 + fib_0
    fib_3 = fib_2 + fib_1
    fib_4 = fib_3 + fib_2
    return fib_4

On the first line we see the def keyword followed by the function's name, which is fifth_fibonacci. This is a fitting name because this function computes the fifth number in the Fibonacci sequence, which we learned about at the end of the previous chapter. Then we have a pair of empty parentheses. If this function took an input then we would put the input inside the parentheses, but it doesn't have an input so the parentheses are empty. On the next few lines we see how the function works. It computes the fifth Fibonacci number, using basically the same calculation we did at the end of the previous chapter. Finally we see the return keyword, which specifies the output of the function. In particular this function's output is fib_4, which has been assigned to the value of the fifth number in the Fibonacci sequence.

Here's an important point. I'll mention it again later because it's so gosh darn essential, but pay attention anyways. After running the code above, we have not computed the fifth Fibonacci number. We have made a function, called fifth_fibonacci. If we use this function, then it will compute the fifth Fibonacci number. But if we never use this function, then the code inside it will never get executed and we will never compute the fifth Fibonacci number.

Also keep in mind that every function has an output. If you don't specify an output with the keyword return, then Python will make your function output None by default. For instance, these next three functions are all completely the same and they all output None:

def example_1():
    variable = 10
    return None

def example_2():
    variable = 10
    return

def example_3():
    variable = 10

Python also stops reading as soon as it sees the keyword return. The next two functions are also completely identical, as far as Python is concerned:

def example_1():
    return 10
    return 20

def example_2():
    return 10

That's because it doesn't really make sense to keep reading example_1, after you already know it's output is the number 10.

Pointers

Recall from the previous chapter, I said variables can only be assigned to one of a few different primitive types: integers, floats, strings, booleans, or None. Well there's a problem with that. A function is not any of those primitive types! So how's it even possible for a variable to be assigned to a function? Short answer: It isn't. That's literally impossible. But we can do something very close.

Instead of binding the variable to a function, you can bind it to an arrow (a.k.a. pointer) that points at a function. This is a subtle difference but it will become very important later on. Take a look at this pyagram, for the fifth_fibonacci function we defined in the previous section. For comparison I've also included a variable called meaning_of_life, bound to 42.

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function fifth_fibonacci() [p = global]
<div style="text-align: right"><span>function fifth_fibonacci() [p = global]</span></div>

First look at the variable called meaning_of_life. It's bound to 42, so the number 42 appears in the little box next to it. Now compare this against the variable named fifth_fibonacci. Notice how the little box does not contain a function! It contains the beginning of an arrow. So just like meaning_of_life is bound to 42, we can see fifth_fibonacci is bound to the beginning of an arrow. That arrow can point wherever it wants. In this example it points at a function, but it could just as feasibly point at a list, or a dog, or whatever else you like. Whenever you want to bind a variable to something that isn't a primitive type, you can just bind the variable to an arrow that points at the thing you want.

Once this makes sense, look at how we represent the function in the pyagram above. First we write "function", to specify that it is a function. Then we write the name of the function, along with the parentheses. If fifth_fibonacci had any inputs, then those would go inside the parentheses. We also write "[p = global]", which tells us the global frame is the parent of the function named fifth_fibonacci. Every function has a parent frame, and the parent frame is always the frame where the function was defined. In other words, it's the frame you were in when you saw the def statement. For the next few chapters, every function we work with will be defined in the global frame, so all of them will have the global frame for their parent.

In summary, you can't bind a variable to a function. Instead you bind it to an arrow, also called a pointer, that points at the function. We'll use this same trick later on, to refer to lists and other things that aren't primitives. It may seem kind of nit-picky and irrelevant now, but this is going to be a big topic later on. Please check that you really understand this section, before you keep reading.

Working with pointers

Really, working with pointers is no different from working with any of the primitive types we learned about in the previous chapter. You just have to keep in mind that the variable is not bound to the function. It's bound to a pointer at the function. Let's see some examples.

Consider the code below, and the corresponding pyagram:

x = 5
y = x

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First we bind x to 5. Then we bind y to the value of x, which is 5. So we copy 5 from x into y, and both variables end up with the value 5. This should just be review, if you read the previous chapter.

Now consider this next example, and the pyagram that goes along with it:

def f():
    return 5

g = f

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First we bind f to a pointer at a function. Then we bind g to the value of f, which is a pointer. So we copy the pointer from f into g, and both variables end up bound to a pointer. Notice we did not copy the function! We only copied the pointer to the function. There's still only one function, but there are now two pointers to it.

Be careful though! It can get tricky. Take a look at this example:

def f():
    return 5

g = f

def f():
    return 10

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function f() [p = global]
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This returns 5.
This returns 5.
This returns 10.
This returns 10.

First we define f, and it gets bound to a pointer at a function whose output is 5. Then we see g = f and g becomes a pointer at the same function. Last, we reassign f to pointer at a function whose output is 10. g still points at the original function, but f points to the new one.

If you then evaluated the variable g in the Python interpreter, you'd see something like this:

>>> g
<function f at 0x101e736a8>

This is just Python's way of telling you the variable g evaluates to a pointer at a function named f. The big scary hexadecimal number is telling you where to find that pointer in your computer, but you don't have to worry about that until you take a computer architecture course.

Here's another thing you should be aware of. A function can exist, even if there's no pointer to it. For instance:

def f():
    return 5

f = 5

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The def statement did two things. It created a function, and it bound the variable f to a pointer at that function. Then we decided to reassign f to the number 5, so the previous value of f — the pointer — disappeared. We never got rid of the function we created. We can't use it anymore because there are no other variables pointing to it, but it's still there.

Calling functions

Now that we have talked about how to define functions, let's talk about actually using them. We refer to this as calling the function.

Syntax for calling functions

Consider our example fifth_fibonacci from earlier:

def fifth_fibonacci():
    fib_0 = 0
    fib_1 = 1
    fib_2 = fib_1 + fib_0
    fib_3 = fib_2 + fib_1
    fib_4 = fib_3 + fib_2
    return fib_4

fifth_fibonacci is just a function whose output is the fifth Fibonacci number. Writing fifth_fibonacci doesn't actually execute any of the code above. In order to do that, you need to write fifth_fibonacci() with the parentheses at the end. This tells you to go actually go through the code above and compute the fifth Fibonacci number. The difference is that fifth_fibonacci is a function, whereas fifth_fibonacci() is a call to that function. The function call returns a value, which you can use or assign to a variable. For instance, this code will evaluate the function call fifth_fibonacci() and bind the resulting integer to a variable called fifth_fibonacci_number:

fifth_fibonacci_number = fifth_fibonacci()

Now let's examine a slightly more nuanced example. Read this code, and keep in mind that dividing by 0 will cause an error to occur in your program:

def divide_by_zero():
    return 1 / 0

function = divide_by_zero
function_call = divide_by_zero()

Remember, the code inside divide_by_zero doesn't happen until we call it. So we actually don't get an error until the very last line in the code above. The def is okay, because that just makes a function and binds the variable divide_by_zero to a pointer at that function. We haven't yet used the function, which means we haven't yet tried to divide by zero. It's also okay to say function = divide_by_zero, because that's just binding the variable function to a pointer. We only encounter an issue when we see function_call = divide_by_zero(), because calling the function causes us to go and execute the code inside it. At last, we attempt to divide by zero and this causes an error to occur in our program.

It's very important to recognize the difference between a function and a function call. When you're just talking about a function, you don't execute the code inside it. That only happens when you're calling the function. It can be easy to confuse the two, especially if you're just starting out, so be extra vigilant about this.

Functions with parameters

So far we've only seen functions that return the same thing every time you call them. Most of the time, though, you'll write functions that return something different depending on a few input variables called parameters. When you want your function to take input parameters, you write them inside the parentheses after the function's name. Then, within the function, you can refer to these parameters just like you would refer to any normal variable. For example, here's a function that takes one parameter called number and outputs its square:

def square(number):
    return number ** 2

Whenever you call a function that requires parameters, you need to provide a concrete value for each parameter. These values are called arguments.

>>> square(number)
Error: 'number' not defined
>>> square(5)
25

Technically the parameter is the variable itself, whereas the argument is the value given to the variable. So in the above example, number is a parameter and 5 is the corresponding argument. But terminology is pretty pointless so there's no need to be pedantic about this, and don't worry about memorizing it. You might also hear some folks refer to the signature of a function. That's the name of the function, including all its parameters — for instance square(number).

Here's another example of calling a function that has parameters, this time taking the average of 3 numbers:

>>> def average(x, y, z):
...     return (x + y + z) / 3
...
>>> z = 8
>>> average(4, z, z - 2)
6.0

The order of evaluation

Before we go any further, we should take the time to understand how function calls actually work. Now is a good time to review the pyagrams we learned about in the previous chapter, because we're about to see them get a little bit more complicated.

Imagine I asked you to compute f(x). You would probably have a few questions. First of all, what does f do? And second, what's x? It's kind of impossible to tell me what f(x) is, when you don't know either of these things. It's similar when Python sees a function call, like average(4, z, z - 2). First of all Python needs to know what average does, and second it needs to process 4, z, and z - 2. It's only possible for Python to compute average(4, z, z - 2) after it knows all those things. So when you see a function call, like average(4, z, z - 2), two things happen:

  1. Python evaluates everything in your expression. It evaluates average to find out what it does. Then it evaluates 4, z, and z - 2 because it's going to need to know what they are too.
  2. Now that it knows all the constituent parts of your function call, Python has to actually perform the computation. Line by line, it executes the code inside average until it reaches the return statement.

Let's see how to do that whole procedure in a pyagram. We'll use this code as an example:

def average(x, y, z):
    total = x + y + z
    return total / 3

z = 8
avg = average(4, z, z - 2)

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As always, we start out with the global frame.

First we define the function average.

Then we assign z to the integer 8.

Finally we arrive at the last line of code, avg = average(4, z, z - 2). Unfortunately we don't know the value of average(4, z, z - 2) so we don't know what to bind to avg yet. So now we have to figure out what the function call average(4, z, z - 2) evaluates to.

In a pyagram, every function call corresponds to something called a flag (because it looks like a little flag). It appears just beneath the current frame, which is the global frame in this example, and we write the function call on the flag banner like so.

Then we have to evaluate everything written on the flag banner: average, 4, z, and z - 2. We'll start with average. Keep in mind that when we evaluate a variable, we need to look it up in the frame above the flag. In this example that's the global frame, so that's where we'll be looking up average. When we search for average in the global frame, we see it's bound to an arrow. Copy that arrow down into the banner, just beneath where we've written average.

We also have to evaluate the other things written on the flag's banner: 4, z, and z - 2.

  1. 4 directly evaluates to 4.
  2. To get z we follow the same steps we used to evaluate average. We look it up in the global frame and find out it's bound to 8.
  3. To get z - 2 it's a similar process. Look up z in the global frame, subtract 2, and we get 6.

Write these values underneath their respective expressions in the flag's banner. Check that you understand everything up to this point.

Now that we know all the constituent parts of our function call, we have to actually perform the computation. Importantly, this doesn't happen in the global frame. Instead we're going to open up a new frame, explicitly dedicated to the code that gets executed due to our function call average(4, z, z - 2). We'll call it "frame 1" because it's the first frame other than the global frame. We also write "average", to make it clear that's the name of the function we're calling.

Last, we write "p = [global]" to indicate frame 1's parent is the global frame. Every frame has a parent frame, similar to how every function has a parent frame. The parent frame for frame 1 is the global frame because frame 1 is for a call to average, and average's parent frame is global. The parent of a frame should always correspond to the parent of the function that's being called.

Now, within frame 1, the first step is to bind the parameters to their respective arguments. This step is easy enough, because we can just copy down the stuff we wrote in the flag banner a few moments ago. Notice how the variable named z in frame 1 is different from the variable named z in the global frame.

At last it's time to evaluate the code inside average. First we have total = x + y + z, which we represent in the same way that we learned in the previous chapter. It gets bound to 18 within frame 1.

Then we have the return statement, which corresponds to a box labeled return at the bottom of the frame. You will always include this box at the end of every frame except the global frame, to indicate the output of the function call you're evaluating.

At this point, we're done evaluating average(4, z, z - 2), marking the end of both frame 1 and the flag. We can go back to the global frame, having discovered the function call evaluates to 6.0. Now we are able to finish the last line in the code above, by binding avg to its proper value.

This is hella important, so double-check that you thoroughly understand everything above. For future reference, here's the procedure. You should apply it whenever you come across a function call.

  1. Make a flag underneath the current frame. Write the entire function call on the flag banner.
  2. Evaluate the function and all its inputs.
    • First look up the function's name in the frame above the flag. If you don't find it there, search in the parent frame. Then the parent frame's parent frame, and so on, until you find it. If you don't even find it after searching the global frame, then Python will throw an error. Whatever you find it bound to, copy that value down into the banner, just beneath the function's name.
    • Then evaluate all the arguments to the function call, and write their respective values beneath them in the flag banner. If any arguments involve variables, then look up those variables using the same procedure as in the bullet point above. If any arguments involve function calls, then pause, start a new flag inside this one, and apply this entire procedure to evaluate that function call before you proceed.
  3. Perform the computation.
    • Make a new frame inside the flag and give it a frame number. Then write the name and parent frame of the function being called. To get this information, you should simply have to follow the arrow that you copied down to the flag banner under the function's name.
    • Inside the frame, bind all the parameters to their respective arguments. For this step, you should only have to refer to the values you wrote in the flag banner.
    • Then walk through the code that gets executed, line by line. If you come across another function call, then apply this entire procedure to evaluate it.
    • Once you reach the return statement, make a corresponding return box at the end of the frame. In that box, write the output from the function call. This marks the end of both the frame and the flag.
    • Now that you know the output of the function call, return to where you were before and continue from there. This will be the frame above the flag that you just finished.

Use this procedure as a guide while you're just learning these concepts, but don't become reliant on it.

Also notice in this procedure that we evaluate all the arguments before looking at the code within the function. This is still the case, even if the function never uses its parameters! So, if the provided arguments cause an error then the function never gets executed at all.

>>> def five(x, y):
...     return 5
...
>>> five(3, 1 / 0)
Error

Again, this whole section is super duper important so make sure you really get it. We'll see some more complicated examples in the next section.

Practice: difference from the average

This stuff can get pretty tricky, so let's practice by drawing a few pyagrams. In particular, we'll look at three equivalent ways of calculating the difference between a number n and the average of three numbers x, y, and z. Even though all three methods produce the same result, they work differently and so they correspond to different pyagrams.

Here's our first method for calculating the difference between n and the average of x, y, and z. At the end, the result is bound to the variable diff_from_avg.

def difference(n, avg):
    return n - avg

avg = average(1, 2, 3)
diff_from_avg = difference(10, avg)

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function difference(n, avg) [p = global]
function difference(n, avg) [p = global]
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function average(x, y, z) [p = global]
function average(x, y, z) [p = global]
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function difference(n, avg) [p = global]
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The first part is very similar to what we did in the previous section, so we'll skip to the bit right after we've assigned avg in the global frame to the result of the function call average(1, 2, 3).

Now it's time for difference(10, avg). The function call is in the global frame, so we should open the new flag at the same indentation.

Then we have to fill in the flag banner with the values of difference, 10 and avg. First let's evaluate difference. We look it up in the global frame and find that it's bound to an arrow. We write that arrow underneath difference in the flag banner.

10 just evaluates to 10, and avg evaluates to 2.0 because that's its value in the global frame.

Next we open frame 2 for our function call. It's parent is the global frame because that's the parent of difference, which is the function we're calling.

Inside frame 2, the first step is to bind the parameters to their respective arguments.

Finally, within frame 2, we execute the code inside difference. It seems to return the float 8.0.

Having determined the value of the function call, we can go back to the global frame. There we bind the result to diff_from_avg. This completes the pyagram.

Here's another method for computing the difference between n and the average of x, y, and z. Instead of two distinct function calls, it uses a nested function call.

def difference(n, avg):
    return n - avg

diff_from_avg = difference(10, average(1, 2, 3))

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global
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function average(x, y, z) [p = global]
function average(x, y, z) [p = global]
average
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difference
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<div style="text-align: right"><br></div>

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function difference(n, avg) [p = global]
function difference(n, avg) [p = global]
frame 1
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average(1, 2, 3)
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global
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<div style="text-align: right"><br></div>
function average(x, y, z) [p = global]
function average(x, y, z) [p = global]
average
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difference
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<div style="text-align: right"><br></div>

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function difference(n, avg) [p = global]
function difference(n, avg) [p = global]
frame 1
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global
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<div style="text-align: right"><br></div>
function average(x, y, z) [p = global]
function average(x, y, z) [p = global]
average
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<div style="text-align: right"><br></div>

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function difference(n, avg) [p = global]
function difference(n, avg) [p = global]
frame 1
average
[p = global]
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frame 2
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global
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<div style="text-align: right"><br></div>
function average(x, y, z) [p = global]
function average(x, y, z) [p = global]
average
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difference
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<div style="text-align: right"><br></div>

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function difference(n, avg) [p = global]
function difference(n, avg) [p = global]
frame 1
average
[p = global]
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<div style="text-align: right"><br></div>
function average(x, y, z) [p = global]
function average(x, y, z) [p = global]
average
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<div style="text-align: right"><br></div>

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function difference(n, avg) [p = global]
function difference(n, avg) [p = global]
frame 1
average
[p = global]
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8.0
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<div style="text-align: right"><br></div>
function average(x, y, z) [p = global]
function average(x, y, z) [p = global]
average
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difference
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<div style="text-align: right"><br></div>
8.0
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function difference(n, avg) [p = global]
function difference(n, avg) [p = global]
frame 1
average
[p = global]
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We'll start off in the global frame, with average already defined.

The first thing we do is define the function difference.

Then we encounter the line diff_from_avg = difference(10, average(1, 2, 3)). But we don't know the value of difference(10, average(1, 2, 3)), so we can't bind diff_from_avg yet.

We will have to make a flag for difference(10, average(1, 2, 3)).

Then we have to fill in the flag banner. difference evaluates to an arrow, as we can see by looking it up in the global frame.

10 directly evaluates to the integer 10. Almost done.

In order to finish filling in the banner we need to know average(1, 2, 3). Referring to the procedure from the previous section, it looks like we just have to start another flag inside this one.

Time to fill in the banner for the flag we just created. To determine the value of average we look it up in the global frame, where we find it's bound to an arrow.

We also evaluate 1, 2, and 3.

Now that the flag for average(1, 2, 3) is done, we open a frame. (Remember, this is all in the procedure from the previous section, above.) It's the first frame other than global, so we'll call it frame 1. It's parent is the global frame because it corresponds to a call to average, and the parent of average is the global frame.

Bind the parameters to their respective arguments.

Then we proceed to do the computation, referring to the definition of average we used before.

It outputs the float 2.0.

Finally, we know what average(1, 2, 3) evaluates to. Now we can finish filling in the banner for the first flag we made.

With the flag banner complete, we are all set to make frame 2 for difference(10, 2.0).

Bind the parameters to their respective arguments.

difference returns n - avg, which evaluates to 8.0.

Now at last we have discovered difference(10, average(1, 2, 3)) is 8.0. We can go back to where we left off in the global frame, and bind diff_from_avg to its appropriate value.

Here's our third and last method of computing the difference between n and the average of x, y, and z. It has a function call inside get_difference_from_avg.

def get_diff_from_avg(n, x, y, z):
    avg = average(x, y, z)
    return n - avg

diff_from_avg = get_diff_from_avg(10, 1, 2, 3)

global
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<div style="text-align: right"><br></div>
function average(x, y, z) [p = global]
function average(x, y, z) [p = global]
average
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function average(x, y, z) [p = global]
function average(x, y, z) [p = global]
average
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get_diff_from_avg
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<div style="text-align: right"><br></div>
function get_diff_from_avg(n, x, y, z) [p = global]
function get_diff_from_avg(n, x, y, z) [p = global]
global
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diff_from_avg
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<div style="text-align: right"><br></div>
function average(x, y, z) [p = global]
function average(x, y, z) [p = global]