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
$$f(x) = \cos(x) e^{-\frac{x^2}{5}}.$$
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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!!!!
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def f(x):
z = cos(x) * exp(-(x**2/5.0))
return z
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def g(x):
return cos(x)**2 + exp(x)
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.
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z = g(6)
y = f(4)
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y
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z
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:
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f(g(g(f(3))))
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def noReturnFct(x,y):
print x + y
return 'My grandma'
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z = noReturnFct(8,9)
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z
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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)
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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.
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f(4)
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f(4, mu=3)
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f(mu=3, 4)
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f(mu=3, x=4)
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