Multiplying expressions within a list of pure functions

Let's say I have a list of pure functions that are nice algebraic expressions: say

l = {(#1 - 1)&, (#1^2 + #1)&, (#1^3 - 1)&}

What's an easy way to get a pure function that will give the product of these expressions? For example, in the above I'd like to get a function f with

f = (#1 - 1)(#1^2 + #1)(#1^3 - 1)&

I've tried (Times @@ Identity @@@ l)&, but this just gives (#1 - 1)(#1^2 + #1)(#1^3 - 1). Essentially, it seems my difficulty is converting a List of Function expressions to a Function[Times[...]] expression, and I can't see how to "strip off" the functions without messing up the referencing of the Slot[1] expressions inside.

Thanks for any help.

Answer

Mapping First over your list will strip off the Function head. Then multiply them together with Apply[Times... and finally Apply[Function... makes the result a pure function. Use the final Apply so the argument to Function is evaluated first as Function has the attribute HoldAll.

Apply[Function,{Apply[Times, Map[First,l]]}]

Gives...

(-1+#1) (#1+#1^2) (-1+#1^3)&

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