Question

Find f(x) if g(f(x))=x and g(x)=(x+1)/(x-1)​

Answer 1

f(x) = (x +1)/(x -1)

Explanation:

Let f(x) = y. We want to find y such that ...

g(y) = x

(y +1)/(y -1) = x

y +1 = xy -x . . . . . . multiply by y-1

y -xy = -x -1 . . . . . . add -1 -xy

y(x -1) = x +1 . . . . . . multiply by -1 and factor out y

y = (x +1)/(x -1) . . . . divide by the coefficient of y

Then ...

f(x) = (x +1)/(x -1)

_____

Check

g(f(x)) = ((x +1)/(x -1) +1)/((x +1)/(x -1) -1)

= ((x +1 +x -1)/(x -1)) / ((x +1 -x +1)/(x -1))

= 2x/2

= x