Quick questions on Inverse relations, the horizontal line test and inverse functions

A relation is just a set of ordered pairs $(x, y)$. Its inverse relation is the set of all the reversed pairs $(y, x)$: whatever the relation sent $x$ to, the inverse sends back. So the way to get the equation of the inverse is to swap $x$ and $y$ everywhere in the equation (and in any restriction), then, if possible, solve for $y$ to get a rule.

The swap-and-solve method shines on a rational function, where the algebra is the whole task. Take $f(x) = \dfrac{2x + 3}{x - 1}$. Swap: $x = \dfrac{2y + 3}{y - 1}$. Multiply out: $x(y - 1) = 2y + 3$, so $xy - x = 2y + 3$.

When a function is many-to-one, its inverse is not a function, but you can often restrict the domain to a piece on which the function is one-to-one, and that piece has an inverse function. The classic case is a parabola: the whole of $f(x) = x^2 - 3$ fails the horizontal line test, but the right-hand branch $x \ge 0$ is increasing and one-to-one.

A few functions are their own inverse: $f^{-1} = f$, equivalently $f(f(x)) = x$. Their graphs are unchanged by reflection in $y = x$ (they are symmetric in that line). The simplest is $f(x) = \dfrac{6}{x}$: $f(f(x)) = \dfrac{6}{6/x} = x$. More generally, a rational function $\dfrac{ax + b}{cx + d}$ is self-inverse exactly when $a + d = 0$.