Let’s say f:D→R is an injective function on some domain where it is also differentiable. For a real function, i.e. D⊂R,R⊂R, is it possible that f′(x)≡f−1(x)?
Intuitively speaking,
I suspect that this is not possible, but I can’t provide a reasonable proof since I knowvery littlenothing about functional analysis. Can anyone provide a (counter)example or prove that such function does not exist?
Answer
It is possible! Here is an example on the domain D=[0,∞):
f(x)=(√5−12)(√5−1)/2x(√5+1)/2.
I found this by supposing that f(x) had the form axb, setting the derivative equal to the inverse function, and solving for a and b.
Attribution
Source : Link , Question Author : polfosol , Answer Author : Greg Martin
