If f(x)=sqrt(1/(x-2)), find f^(-1)(x)=______ and (f^(-1)'(x)=______
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If f(x)=sqrt(1/(x-2)), find f^(-1)(x)=______ and (f^(-1)'(x)=______

[From: ] [author: ] [Date: 11-10-12] [Hit: ]
.. restricted to the domain y>0 so that the domain of f^-1 is identical to the range of f.If you want to change y to x in that definition of f^-1, go ahead, but just remember its not the same x as in f(x).......
I don't like using x as the dummy variable for f^-1, particularly when the domain and range of f are not the same. f has the domain x>2, and the range f(x)>0. Write this as:

y = f(x) = sqrt(1/(x - 2)) .... where x>2 is the domain and y>0 is the range
y² = 1/(x - 2)
x - 2 = 1/y²
x = 2 + 1/y² = (2y² + 1)/y²

So f^-1(y) = (2y² + 1)/y² .... restricted to the domain y>0 so that the domain of f^-1 is identical to the range of f.

If you want to change y to x in that definition of f^-1, go ahead, but just remember it's not the same x as in f(x).

Edit: Oh yeah, the derivative. I should have left it in the first form for that, where d/dy f^-1(y) = -2/y^3 is obvious.
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