Checking if input is a function

Question

I would like to write a program to find the leading coefficient of a given polynomial. (Sounds easy, right?) I'd like to be able to handle the format x^2-x+7 (with any formal variable x) as well as the format #^2-#+7&. I would also like to be able to reject/error on inputs which are not univariate polynomials.

I thought the function Variables would make this easy, but it turns out that

Variables[#^2-#+7&]

gives

{#1^2 - #1 + 7 &}

which isn't helpful. My current plan:

Check if the input is a function; if so, substitute f[\[FormalX]] for f.
Check if Variables[f] has length other than 1; if so, fail.
Define x as Variables[f][[1]] and return Coefficient[f,x,Exponent[f,x]].

But I'm stuck on the first step. Is there a good way to do this? Is there a better approach to what I'm trying to do?

Does checking the head work, i.e. yourFunction[x_Function]:= do something to x.? — N.J.Evans, Apr 25 '16 at 16:39
@J.M. That's interesting -- is there a way to convert an anonymous function to this form automatically? — Charles, Apr 25 '16 at 16:41
also Variables @@ (#^2 - # + 7 &) or Variables @@ Function[#^2 - # + 7] — kglr, Apr 25 '16 at 17:53

score 2 · Accepted Answer · edited Apr 13 '17 at 12:55

2

lead[f_Function ] := lead[f@\[FormalX]]
lead[f_?PolynomialQ] /; (Length[(v = Variables[f])] == 1) := 
 Last@CoefficientList[f, v[[1]]]
lead[2 x^2 - x + 7 ]
lead[ 3 #^2 - # + 7 &]

2

3

Note this breaks if you supply a pure function with more than one slot. See here if you need to deal with that: How can I get the number of slots in Function?

edited Apr 13 '17 at 12:55

Community

1

answered Apr 25 '16 at 20:51

george2079

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Checking if input is a function

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