Inverse Functions

What AP Precalc asks here

An inverse function $f^{-1}$ undoes what $f$ does: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. The inverse exists if and only if $f$ is one-to-one (passes the horizontal line test). Algebraically: write $y = f(x)$, swap $x$ and $y$, solve for $y$. Graphically, the inverse is the reflection across $y = x$; domain and range swap.

Write $y = f(x)$, swap $x$ and $y$, then solve for the new $y$.

Compute $f(g(x))$ and $g(f(x))$. Both must simplify to $x$.

Swap coordinates of points; swap domain and range; reflect graphs across $y = x$.