Is the graph of $y=\frac{k}{x}$ a hyperbola? Is the graph of the following inverse relation a hyperbola?$$y=\frac kx$$ If yes, is it the only kind of hyperbola whose equation is an explicit function?

Yes, $$y=\frac kx$$ is a hyperbola. In fact, it is a type of _rectangular hyperbola_ which you can read more about here.

This means that a function $y(x)$ that takes the form $$y-k=\frac {k}{x-h}$$ is also hyperbola.

~~I suppose that is the only type of hyperbolas that is a function since any rotation of this type of hyperbolas will immediately cause the plot to fail the vertical line test.~~

Thanks to @Blue:

> Any (non-degenerate) hyperbola with a vertical asymptote is the graph of a function. Rectangularity is not a requirement.