What is the inverse of the function f(x) = (x + 2)/(x - 2)?

To find the inverse of the function ( f(x) = \frac{x + 2}{x - 2} ), one approach is to replace ( f(x) ) with ( y ), leading to the equation ( y = \frac{x + 2}{x - 2} ). The next step would be to switch the roles of ( x ) and ( y ) to find the inverse, resulting in ( x = \frac{y + 2}{y - 2} ).

From here, to solve for ( y ), you can cross-multiply to eliminate the fraction, which gives you the equation ( x(y - 2) = y + 2 ). Expanding this results in ( xy - 2x = y + 2 ). Next, re-arranging the terms helps isolate ( y ):

1. Move all terms involving ( y ) to one side and the rest to the other: ( xy - y = 2 + 2x ).

2. Factor out ( y ) on the left side: ( y(x - 1) = 2 + 2x ).

3. Finally, solve for ( y ): (