Python provides many built-in functions, but as we start writing more code ourselves, it will be essential that we be able to create our own functions specific to the problem we are solving. In this section, we’ll learn how to define our own functions in Python.
First, let’s recall how we define a function in mathematics. We first specify the function name, domain, and codomain, for example, \(f: \R \to \R.\) Then, we write the function header and body, usually in a single line: for example, \(f(x) = x^2\). We do this so often in mathematics that we often take parts of this for granted, for example leaving out the domain/codomain specification, and usually choosing \(f\) as the function name and \(x\) as the parameter name. However, the functions we’ll implement in Python are much more diverse, and so it will be important to be explicit in every part of this process.
Here is the complete definition of a “squaring” function in Python. Take a moment to read through the whole definition, and then continue reading to learn about this definition’s different parts.
def square(x: float) -> float:
"""Return x squared.
>>> square(3.0)
9.0
>>> square(2.5)
6.25
"""
return x ** 2This function definition is the most complex form of Python code we’ve seen so far, so let’s break this down part by part.
The first line, def square(x: float) -> float: is
called the function header. Its purpose is to convey
the following pieces of information:
square).x and type
float.->. In this example, the function
return type is
float. For square, its parameter and return type
are the same, but this doesn’t have to be the case in
general.The syntax for a function header for a unary function is:
def <function_name>(<parameter_name>: <parameter_type>) -> <return_type>:Compared to our mathematical version, there are two main differences.
First, we chose the name square rather than f
as the function name; in Python, we will always pick descriptive names
for our functions rather than relying on the conventional “\(f\)”. And second, we use data types to
specify the function domain and codomain: the code x: float
specifies that the parameter x must be a float
value, and the code -> float specifies that this
function always returns a float value.
We can express this restriction in an analogous way to \(\R \to \R\) by writing
float -> float; we call float -> float
the type contract of the square
function.
The next seven lines, which start and end with triple-quotes
("""), is called the function docstring.
This is another way of writing a comment in Python: text that is meant
to be read by humans, but not executed as Python code. The goal of the
function docstring is to communicate what the function does. The
function docstring is indented inside the function header, as a visual
indicator that it is part of the overall function
definition. Unlike many other programming languages, this kind of
indentation in Python is mandatory rather than merely
recommended. Python’s designers felt strongly enough about indentation
improving readability of Python programs that they put indentation
requirements like this into the language itself.
The first part of the docstring, Return x squared., is
an English description of the function. The second part might look a bit
funny at first, since it seems like Python
code:Or more precisely, it looks like the Python
console!
>>> square(3.0)
9.0
>>> square(2.5)
6.25This part of the docstring shows example uses of the function, just
like the examples we showed of built-in functions in the previous
section. You can read the first example literally as “evaluating
square(3.0) in the Python console returns 9.0”
and the second as “evaluating square(2.5) in the Python
console returns 6.25”. These examples are called
doctest examples, for a reason we’ll see in a future
section. While an English description may technically be enough to
specify the function’s behaviour, doctest examples are invaluable for
aiding understanding of the function behaviour (which is why we use them
in teaching as well!).
The final line, return x ** 2, is called the
body of the function, and is the code that is executed
when the function is called. Like the function docstring, the function
body is also indented so that it is “inside” the function
definition.
This code uses another keyword, return, which signals a
new kind of statement: the return statement, which has
the form:
return <expression>When the Python interpreter executes a return statement, the following happens:
<expression> is evaluated, producing a
value.Okay, so that’s a function definition. But how do we actually use such a definition in Python? While it is possible to define functions directly in the Python console, this isn’t a good approach: every time we restart the Python console, we lose all our previous functions definitions and have to re-type them.
So instead, we save functions in files so that we can reuse them
across multiple sessions in the Python console (and in other files, as
we’ll see later). For example, we can create a new file called
my_functions.py, and write our function definition in that
file: We strongly encourage you to follow along with this in
PyCharm!
def square(x: float) -> float:
"""Return x squared.
>>> square(3.0)
9.0
>>> square(2.5)
6.25
"""
return x ** 2Then after saving the file, we can right-click on the file in PyCharm
and select “Run File in Python Console”. This will start the Python
console and run our file, which then allows us to call our function
square just like any built-in function:
>>> square(3.0)
9.0
helpIt turns out that function docstrings aren’t just regular comments—they are are read and stored by the Python interpreter at each function definition. You might wonder why, though, if these docstrings are just text that explain what a function is supposed to do.
Here is one very cool reason: a function’s docstring is part of
the documentation that is displayed when a user calls help
on that function. So when we used help to look at some
of the built-in functions in the previous section, the Python
interpreter was actually showing us the docstrings of those functions!
To demonstrate this, let’s call help on our
square function in the Python console:
>>> help(square)
Help on function square in module my_functions:
square(x: float) -> float
Return x squared.
>>> square(3.0)
9.0
>>> square(2.5)
6.25Neat!
Let’s now look at a more complex example that will illustrate a function definition that takes in more than one parameter.
Recall the distance formula from Section 1.6 to calculate the distance between two points \((x_1, y_1), (x_2, y_2)\) in the Cartesian plane: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
We’ll now write a function in Python that calculates this formula.
This function will take two inputs, where each input is a
list of two floats, representing the \(x\)- and \(y\)-coordinates of each point. When we
define a function with multiple parameters, we write the name and type
of each parameter using the same format we saw earlier, with parameters
separated by commas from each other. Here is the function header and
docstring:
def calculate_distance(p1: list, p2: list) -> float:
"""Return the distance between points p1 and p2.
p1 and p2 are lists of the form [x, y], where the x- and y-coordinates are points.
>>> calculate_distance([0, 0], [3.0, 4.0])
5.0
"""In order to use the above formula, we need to extract the coordinates
from each point. This is a good reminder of list indexing,
and also introduces a new concept: function bodies can consist of more
than one statement.
# The start of the body of calculate_distance
x1 = p1[0]
y1 = p1[1]
x2 = p2[0]
y2 = p2[1]Now that we have the four coordinates, we can apply the above formula and return the result.
# Continuing the function body
return ((x2 - x1) ** 2 + (y2 - y1) ** 2) ** 0.5Putting this all together, we have:
def calculate_distance(p1: list, p2: list) -> float:
"""Return the distance between points p1 and p2.
p1 and p2 are lists of the form [x, y], where the x- and y-coordinates are points.
>>> calculate_distance([0, 0], [3.0, 4.0])
5.0
"""
x1 = p1[0]
y1 = p1[1]
x2 = p2[0]
y2 = p2[1]
return ((x2 - x1) ** 2 + (y2 - y1) ** 2) ** 0.5If we save this function in our my_functions.py file
from earlier, we can then “Run File in Python Console” and call this
function in the Python console:
>>> calculate_distance([0, 0], [3.0, 4.0])
5.0Our above function body is perfectly correct, but you might notice
that the ** 2 expressions exactly mimic the body of the
first function we defined in this section: square. And so
we can call our square function inside the body of
calculate_distance:
def calculate_distance(p1: list, p2: list) -> float:
"""Return the distance between points p1 and p2.
p1 and p2 are lists of the form [x, y], where the x- and y-coordinates are points.
>>> calculate_distance([0, 0], [3.0, 4.0])
5.0
"""
x1 = p1[0]
y1 = p1[1]
x2 = p2[0]
y2 = p2[1]
return (square(x2 - x1) + square(y2 - y1)) ** 0.5This example is still relatively small, so you might wonder why it’s
useful to replace the (x2 - x1) ** 2 with
square(x2 - x1); isn’t that roughly the same amount of
typing?
That’s true in this case, but we wanted to use the example to illustrate a more general principle: it is possible to call a function from inside the body of another function. This is a very useful principle that comes in handy when our programs grow larger and more complex: we’ll see how we can split up a complex block of code into separate functions that call each other, enabling us to better organize our code. We’ll explore this idea in more detail, and other principles of good function and program design, throughout this course.
We’ll wrap up this section with an introduction to some of the technical details of function calls in Python. In the previous section, we called built-in functions and took for granted that they worked properly, without worrying about how they work. Now that we’re able to define our own functions, we are ready to fully understand what happens when a function is called.
As an example, suppose we’ve defined square as above,
and then call it in the Python console:
>>> square(2.5)When we press Enter, the Python interpreter evaluates the function call by doing the following:
2.5, and then assign its value to
the function parameter
x. This is analogous to the concept of
substituting the value 2.5 for variable
x that you use for evaluating mathematical function calls.
square function, by doing:
x ** 2, which is 6.25
(since x refers to the value 2.5).6.25 back to the Python console.square(2.5) evaluates to
6.25, and this is displayed on the screen.We saw in the previous chapter that expressions can be combined
together to form larger expressions, for example
(4 * 5) + (3 * 8). Since function calls are a type of
expression, we can also combine them in larger expressions, like we did
in our calculate_distance example.
What does the Python interpreter do, exactly, when we call
square twice in the same expression? For example, let’s
consider typing the following into the Python console and pressing
Enter:
>>> square(2.5) + square(-1.0)In the description below, notice how we’ve duplicated the text from
before to illustrate the similarities between calling
square(2.5) and -square(-1.0).
+ in
left-to-right order, so evaluate square(2.5) first.
2.5, and then assign 2.5 to the
function parameter x.square function, by doing:
x ** 2, which is 6.25
(since x refers to 2.5).6.25 back to the Python console.square(-1.0) to
be evaluated.
-1.0, and then assign -1.0 to the
function parameter x.square function, by doing:
x ** 2, which is 1.0 (since
x refers to -1.0).1.0 back to the Python console.6.25 + 1.0, which evaluates to 7.25. This
value is displayed on the screen.