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> Find the domain of the inverse of the following function:$$f(x) = kx^\frac{1}{2}$$
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The inverse, using some basic algebraic rearranging, can be said to be:

$$f^{-1}(x) = \frac{k^2}{x^2}$$

As evident, the domain of this function seems to be all real numbers. 

However, a property of functions states that,

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> The domain of $f(x)$ is equal to the range of $f^{-1}(x)$, and the domain of $f^{-1}(x)$ is equal to the range of $f(x)$
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We know that the range of $f(x)$ is $x>0$; 

The presumed domain of $f^{-1}(x)$ doesn't equal the range of $f(x)$. Why is the domain of $f^{-1}(x)$ said to be the range of $f(x)$ even when the inverse function doesn't supposedly need to have a restriction on its domain?