How can I write the functional inverse of a function in maple to be used in calculations?
e.g.,
diff(f^(-1)(x),x) = 1/f'(f^(-1)(x))
when I use f^-1(x) or f^(-1)(x) I get the multiplicative inverse rather than the compositional inverse.
I would like the general form for arbitrary or specified functions. If f is not defined above it should return the general form. I'd like to be able to compute symbolic deratives for functions containing inverses without having to specify a specific function.
You can get (some) support for this using the @@ operator, also known at the "repeated function composition operator". See its help page, eg on-line here.
restart;
expr:=f(x);
expr := f(x)
(f@@(-1))(f(x));
x
finv:=f@@(-1);
(-1)
finv := f
finv(f(x));
x
f(finv(x));
(-1)
f((f )(x))
simplify(f(finv(x)));
x
It turns out that the diff and D commands know a bit of how to deal with it.
diff( finv(x), x );
1
----------------
(-1)
D(f)((f )(x))
D(finv)(x);
1
----------------
(-1)
D(f)((f )(x))
convert(%,diff);
1
---------------------------
/ d \|
|--- f(t1)||
\dt1 /| (-1)
|t1 = (f )(x)
You may find the lprint command useful, to see the underlying 1D plaintext code structure of a result which pretty-prints in Maple's so-called 2D Math or 2D Output.
lprint( convert(D(finv)(x),diff) );
1/eval(diff(f(t1),t1),{t1 = `@@`(f,-1)(x)})
As quoted from their site using $f(x)=\frac{x-2}{x-7}$ as an example (scroll to part C): https://www.maplesoft.com/applications/view.aspx?SID=4088&view=html
$$f := x -> (x-2)/(x-7);$$ $$\text{solve}( x = f(y), y );$$ $$g := \text{unapply }(\%, x);$$
There is no general, universal solution to compute the inverse function of a procedure in Maple. A couple of the other answers cover what you are able to do in some cases.
