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 *Functions, Indentation, and Counting*.
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)
Graphics object consisting of 1 graphics primitive
```

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).

```
sage: var('z') # define z to be a variable
z
sage: f(z)
z^2
sage: plot(f(z), 0, 2)
Graphics object consisting of 1 graphics primitive
```

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)
Graphics object consisting of 1 graphics primitive
```

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.

```
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)
Graphics object consisting of 1 graphics primitive
```

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)
Graphics object consisting of 1 graphics primitive
sage: type(sin(x))
<type 'sage.symbolic.expression.Expression'>
sage: plot(sin(x), 0, 2)
Graphics object consisting of 1 graphics primitive
```

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 the symbolic variable `x`
into the function `h`. So, the inequality `x < 2` evaluates to `False` first,
and hence `h(x)` evaluates to `x - 2`. This can be seen with

```
sage: bool(x < 2)
False
sage: h(x)
x - 2
```

Note that here there are two different `x`: the Python variable used to
define the function `h` (which is local to its definition) and the symbolic
variable `x` which is available on startup in Sage.

The solution: don’t use `plot(h(x), 0, 4)`; instead, use

```
sage: plot(h, 0, 4)
Graphics object consisting of 1 graphics primitive
```

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.”

```
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.