Can you plot pure function without specifying variable?

Question

Given a pure function, e.g. f=Sin[#]&, it is possible to plot it with introducing an arbitrary local variable, e.g.,

Plot[f[x],{x,0,1}]

However, the introduction of a variable x seems unnecessary. Is it possible to plot this without specifying a name for the variable? If not, is there a good reason why this functionality doesn't exist?

Answer 1

Yes, you can plot it, but not using Plot. For example, you could map the function over a range of values and then use ListLinePlot:

With[{xmin = 0, xmax = 4π},
 ListLinePlot[f/@Subdivide[##,100],DataRange->{##}]&[xmin,xmax]
]

This uses the new function Subdivide with 100 plot points.

The reason why Plot requires you to specify a dummy variable is that it takes expressions and not functions as its argument. Therefore, the plot variable is not identifiable by a slot, and you need to specify it by naming the plot variable.

Answer 2

There is the function Unique, which is often used to avoid symbol name collisions.

You could use it like this:

Plot[f[#], {#, 0, 10}] &[Unique[]]

Of course, in another sense, it's the most arbitrary that a variable can get (it's just a variable with a name of the form $12345 where 12345 represents some arbitrary number).

Answer 3

The short answer is no, not with Plot, which like many other Mathematica built-in functions that range over a set of values, is designed to follow a standard syntax pattern for which Table may be considered the prototype. Consider Do or even Manipulate$\text{*}$, which was added long after Plot. Both follow this pattern although in the case of Manipulate the semantics are wildly different.

$\text{*}$ See this conference article for a discussion of how Manipulate was deliberately designed to emulate Table. It also has many additional interesting things to say about Manipulate.

Answer 4

As an additional small note to ciao's comment, it's possible to define your own function based on Plot to work with pure functions, e.g.:

plotPure[fun_, d_, args___] := 
    Plot[Evaluate@If[ListQ[fun], Through[fun[x]], fun[x]], {x, d[[1]], d[[2]]}, args];

This way it provides all Plot functionality:

plotPure[{Sin, Sin[#]^3 &}, {-Pi, Pi}, PlotStyle->"Thick"]