.. _section-functions-issues:
Some Common Issues with Functions
=================================
Some aspects of defining functions (e.g., for differentiation or
plotting) can be confusing. In this section we try to address some of
the relevant issues.
Here are several ways to define things which might deserve to be
called "functions":
1. Define a Python function, as described in :ref:`section-functions`.
These functions can be plotted, but not differentiated or integrated.
::
sage: def f(z): return z^2
sage: type(f)
<type 'function'>
sage: f(3)
9
sage: plot(f, 0, 2)
In the last line, note the syntax. Using ``plot(f(z), 0, 2)`` instead
will give an error, because ``z`` is a dummy variable in the
definition of ``f`` and is not defined outside of that
definition. Indeed, just ``f(z)`` returns an error. The following will
work in this case, although in general there are issues and so it
should probably be avoided (see item 4 below).
.. link
::
sage: var('z') # define z to be a variable
z
sage: f(z)
z^2
sage: plot(f(z), 0, 2)
At this point, ``f(z)`` is a symbolic expression, the next item in our
list.
2. Define a "callable symbolic expression". These can be plotted,
differentiated, and integrated.
::
sage: g(x) = x^2
sage: g # g sends x to x^2
x |--> x^2
sage: g(3)
9
sage: Dg = g.derivative(); Dg
x |--> 2*x
sage: Dg(3)
6
sage: type(g)
<type 'sage.symbolic.expression.Expression'>
sage: plot(g, 0, 2)
Note that while ``g`` is a callable symbolic expression, ``g(x)`` is a
related, but different sort of object, which can also be plotted,
differentated, etc., albeit with some issues: see item 5 below for an
illustration.
.. link
::
sage: g(x)
x^2
sage: type(g(x))
<type 'sage.symbolic.expression.Expression'>
sage: g(x).derivative()
2*x
sage: plot(g(x), 0, 2)
3. Use a pre-defined Sage 'calculus function'. These can be plotted,
and with a little help, differentiated, and integrated.
::
sage: type(sin)
<class 'sage.functions.trig.Function_sin'>
sage: plot(sin, 0, 2)
sage: type(sin(x))
<type 'sage.symbolic.expression.Expression'>
sage: plot(sin(x), 0, 2)
By itself, ``sin`` cannot be differentiated, at least not to produce
``cos``.
::
sage: f = sin
sage: f.derivative()
Traceback (most recent call last):
...
AttributeError: ...
Using ``f = sin(x)`` instead of ``sin`` works, but it is probably even
better to use ``f(x) = sin(x)`` to define a callable symbolic
expression.
::
sage: S(x) = sin(x)
sage: S.derivative()
x |--> cos(x)
Here are some common problems, with explanations:
\4. Accidental evaluation.
::
sage: def h(x):
... if x<2:
... return 0
... else:
... return x-2
The issue: ``plot(h(x), 0, 4)`` plots the line `y=x-2`, not the
multi-line function defined by ``h``. The reason? In the command
``plot(h(x), 0, 4)``, first ``h(x)`` is evaluated: this means plugging
``x`` into the function ``h``, which means that ``x<2`` is evaluated.
.. link
::
sage: type(x<2)
<type 'sage.symbolic.expression.Expression'>
When a symbolic equation is evaluated, as in the definition of ``h``,
if it is not obviously true, then it returns False. Thus ``h(x)``
evaluates to ``x-2``, and this is the function that gets plotted.
The solution: don't use ``plot(h(x), 0, 4)``; instead, use
.. link
::
sage: plot(h, 0, 4)
\5. Accidentally producing a constant instead of a function.
::
sage: f = x
sage: g = f.derivative()
sage: g
1
The problem: ``g(3)``, for example, returns an error, saying
"ValueError: the number of arguments must be less than or equal to 0."
.. link
::
sage: type(f)
<type 'sage.symbolic.expression.Expression'>
sage: type(g)
<type 'sage.symbolic.expression.Expression'>
``g`` is not a function, it's a constant, so it has no variables
associated to it, and you can't plug anything into it.
The solution: there are several options.
- Define ``f`` initially to be a symbolic expression.
::
sage: f(x) = x # instead of 'f = x'
sage: g = f.derivative()
sage: g
x |--> 1
sage: g(3)
1
sage: type(g)
<type 'sage.symbolic.expression.Expression'>
- Or with ``f`` as defined originally, define ``g`` to be a symbolic
expression.
::
sage: f = x
sage: g(x) = f.derivative() # instead of 'g = f.derivative()'
sage: g
x |--> 1
sage: g(3)
1
sage: type(g)
<type 'sage.symbolic.expression.Expression'>
- Or with ``f`` and ``g`` as defined originally, specify the variable
for which you are substituting.
::
sage: f = x
sage: g = f.derivative()
sage: g
1
sage: g(x=3) # instead of 'g(3)'
1
Finally, here's one more way to tell the difference between the
derivatives of ``f = x`` and ``f(x) = x``
::
sage: f(x) = x
sage: g = f.derivative()
sage: g.variables() # the variables present in g
()
sage: g.arguments() # the arguments which can be plugged into g
(x,)
sage: f = x
sage: h = f.derivative()
sage: h.variables()
()
sage: h.arguments()
()
As this example has been trying to illustrate, ``h`` accepts no
arguments, and this is why ``h(3)`` returns an error.
