For x ∈ R - {0, 1}, let f1(x) = 1/x, f2(x) = 1 - x and f3(x) = 1/(1 - x) be three given functions. If a function, J(x) satisfies (f2 ◦ J ◦ f1)(x) = f3(x), then J(x) is equal to

The correct option is A 1/xf3(x)
Given f1(x) = 1/x, f2(x) = 1 - x and f3(x) = 1/(1 - x)
(f2 ◦ J ◦ f1)(x) = f3(x)
f2(J(f1(x))) = f3(x)
1 - J(1/x) = 1/(1 - x)
J(1/x) = 1 - 1/(1 - x)
J(1/x) = (1 - x - 1)/(1 - x)
J(1/x) = -x/(1 - x)
Let y = 1/x
x = 1/y
J(y) = -(1/y)/(1 - 1/y)
J(y) = -(1/y)/((y - 1)/y)
J(y) = -1/(y - 1)
J(y) = 1/(1 - y)
J(x) = 1/(1 - x) = f3(x)
Therefore, J(x) = f3(x) = 1/(1 - x)
Also, 1/xf3(x) = (1/x)(1/(1 - x)) = 1/(x(1 - x))
This is not equal to J(x) = 1/(1-x)
However, (f2 ◦ J ◦ f1)(x) = f2(J(f1(x))) = f2(J(1/x)) = 1 - J(1/x) = f3(x) = 1/(1-x)
So, 1 - J(1/x) = 1/(1-x)
J(1/x) = 1 - 1/(1-x) = (1-x-1)/(1-x) = -x/(1-x)
Let z = 1/x, then x = 1/z
J(z) = -1/z / (1 - 1/z) = -1/z / (z-1)/z = -1/(z-1) = 1/(1-z)
Therefore J(x) = 1/(1-x) = f3(x)
1/x f3(x) = 1/(x(1-x)) ≠ f3(x)