# Functions

### Function definition

Definition: A function in Python is the programmatic realization of the idea of a math function $f:A \rightarrow B$, that is, a rule that tells you how to associate any value $a$ in an input set a unique value $b$ in an output set.

Example: Consider the math function $f:\mathbb R\rightarrow \mathbb R$ given by

$$f(x) = \cos(x) e^{-\frac{x^2}{5}}.$$


In :

from math import exp, cos



In Python, one way to define a function is by using the following construct:

def function_name(arg1, arg2, arg3):
instruction 1
instruction 2
instruction 3
return z



The input of the function $f$ is the argements arg1, arg2, arg3, and the output of $f$ is whatever comes after the keyword return.

Remark: If not the function does not have a return instruction, then nothing is returned!!!!



In :

def f(x):
z = cos(x) * exp(-(x**2/5.0))
return z




In :

def g(x):
return cos(x)**2 + exp(x)



### Function call

A function call is an expression of the form

function_name(val1, val2, val3)



where now val1, val2, val3 are concrete values and not parameters. The interpreter evaluates a function call by replacing the function call by the function output.



In :

z = g(6)
y = f(4)




404.350720472
-0.0266439546028




In :

y




In :

z



### Function Compostion

In math, if you have two functions $f:A\rightarrow B$ and $g:B\rightarrow C$, you can compose these functions by appliying first $f$ to and element $a\in A$ and then the second function $g$ to the output $f(a)\in B$. The result is a new function called the composition: $g \circ f(a):= g(f(a))$ yielding the function $g\circ f:A \rightarrow C$.

In Python (and in any other programming languages), one composes function the same way:



In :

f(g(g(f(3))))




Out:

0.00040507051471082436




In :

def noReturnFct(x,y):
print x + y
return 'My grandma'




In :

z = noReturnFct(8,9)




17




In :

z




Out:

'My grandma'




In [ ]:



## Main parameter and options

We can separate the parameter of function in two types:

• main parameters, that are required for the function to work: it's the INPUT

• optional parameter, that are not required, and that customize the function behaviour: they are the OPTIONS

Remark: Optional parameter are not required because they have default values

f(x, option1=value1, option2=value2, etc.)



If the function call is as follows:

f(4)



then, behind the scene, Python calls the function with the default arguments

f(4, value1, value2)


In :

from math import exp

def f(x, mu=1):
return mu**x * exp(mu * x)



The function above represent the family of functions

$$f_\mu (x) = \mu^x\exp(\mu x),$$

where $x$ is the function input and $\mu$ is the family parameter.



In :

f(4)




Out:

54.598150033144236




In :

f(4, mu=3)




Out:

13183138.104939317




In :

f(mu=3, 4)




f(mu=3, 4)
SyntaxError: non-keyword arg after keyword arg




In :

f(mu=3, x=4)




Out:

13183138.104939317