Determine the inverse of the function f(x) = x-2 / x+2 , where x is not equal to -2.

Question

Determine the inverse of the function f(x) =( x-2) /( x+2) , where x is not equal to -2.

 

Answer 1

To find the inverse of a function, just switch x and y: 

 

1. switch x and y:            x = (y-2)/(y+2)

2. Solve for y 

                                     x(y+2) = y-2 

                                     xy +2x = y-2         <--  distribute the x to (y+2) 

                                     xy -(y-2) = -2x      <-- all I am doing in this step is bringing terms that have "y" to the same side. This is crucial, and is the most important step to solving for y 

                                     xy - y +2 = -2x      <-- Distribute the negative to (y-2). Now, we can finish putting all the terms that have "y" on the same side. We are almost done!

                                    y(x-1) = -2x - 2       <-- I group together the terms that contain y and I move the 2 to the other side of the equation. Now, all the terms that have "y" are on the left side, and all the terms that have "x" are on the right side.

                                     y = -2x-2/(x-1)       <-- Now we have solved for y! The problem is technically solved, but we can reduce this to a nicer form.

                                    y = -2(x+1)/(x-1)    <-- Factor out -2 from the -2x-2. Be sure to account for the minus sign in front of the 2. Factoring out -2 from -2x and -2 yields -2(x+1).

Final answer:                 y= -2(x+1)/(x-1)