Question

*Urgent*

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f(x)=2/x and g(x)=2/x

Answer 1

to the risk of sounding redundant.


`\bf \begin{cases} f(x)=\cfrac{2}{x}\\\\ g(x)=\cfrac{2}{x} \end{cases} \\\\\\ f(~~g(x)~~)=\cfrac{2}{g(x)}\implies \cfrac{2}{(2)/(x)}\implies \cfrac{\quad (2)/(1)\quad }{(2)/(x)}\implies \cfrac{2}{1}\cdot \cfrac{x}{2}\implies x \\\\\\ g(~~f(x)~~)=\cfrac{2}{f(x)}\implies \cfrac{2}{(2)/(x)}\implies \cfrac{\quad (2)/(1)\quad }{(2)/(x)}\implies \cfrac{2}{1}\cdot \cfrac{x}{2}\implies x`

Answer 2

Let's find f(g(x)):

f(x) = 2/x. Let's replace the two instances of x with g(x) = 2/x:


2
f( g(x) ) = --------------- = x. Thus, we have shown that f and g are inverses.
(2/x)