how to get the inverse of f(x)=1/√x?

Question

mww · Answer

To find any inverse from an invertible function, you write y = f(x) as x=g(y). In other words, we change the subject for x. 
Then you replace x with y and y with x.
So for $$f(x) = 1/\sqrt{x} $$
$$\sqrt{x} = 1/f(x)$$
$$x = \frac{ 1 }{ f ^{2}(x) }$$

Having achieved finding x in terms of f(x), Now replace x with y and y with x.
$$y = \frac{ 1 }{ x^2 }$$ or $$f ^{-1}(x) = 1/x^2$$
But we're not finished. Our original function only made sense for x > 0 (Domain) and so y > 0 (range).
Inverse functions have both domain and range swapped. So the domain of the inverse function is the same as the range of the original.
I.e. our function is only defined for x > 0. 
$$f ^{-1}(x) = 1/x^2, x >0 $$

It is not always possible to find the rule for the inverse but it is important to note invertible functions are those that are one-one functions. (every function value occurs not more than once).