1. inverse

if f:[1,infinity)-[1,infinity) is defined by f(x)-2^(x(x-1)) then find inverse of f(x)

2. Certainly y = f(x) is strictly increasing in [1,+infty) and

lim_ {x to +infty} f(x) = +infty

This means that

f : [1,+infty) -> [1,+infty)

is bijective, as a consequence there exists f^{ -1 } .

Take log in both sides of y = f(x) , solve the quadratic equation on x and choose the positive branch.

3. There is some problem of 'interpretation', so I suppose that the function is...

y= 2^[x (x-1)] (1)

... so that its inverse function is the solution(s) of the equation...

x^2 - x - log2 y=0 (2)

... that are...

x= 1/2 [1 +/- (1+ 4 log2 y)^(1/2)] (3)

It has to be noted that (3) has two distinct brantches and is defined for y> 2^(- 1/4)...

Kind regards

chi sigma