as you already know, to get the inverse of any expression we start off by doing a quick switcheroo on the variables and then solving for "y", let's do so.
[tex]\stackrel{f(x)}{y}~~ = ~~\cfrac{4x}{x-9}\implies \stackrel{\textit{quick switcheroo}}{x~~ = ~~\cfrac{4y}{y-9}}\implies xy-9x=4y\implies -9x=4y-xy \\\\\\ -9x=y(4-x)\implies \cfrac{-9x}{4-x}=y\implies \cfrac{9x}{x-4}=\stackrel{f^{-1}(x)}{y}[/tex]
how to test algebraically? well, for any (a,b) pair in f(x), its inverse will have another pair but (b,a), hmmm let's plug hmmm say
f(10) = 40, and f⁻¹(40) = 10.
we could plug more values in each, same pair will come up.
as far as testing them graphically, Check the picture below.
