Question

f is a one-to-one function. what is the inverse of this function?

Answer 1

\[g(x)=\frac{ 2+3f(x) }{ 1-f(x) }\]

Answer 2

\[
\text{If } ~h(x) = \frac{2+3x}{1-x}~~\text{ then }~~g(x) = h(f(x)) \\
\]We can find inverse of h(x) and then find the inverse of g(x).

Answer 3

Very clever !!!

Answer 4

y = (2+3x) / (1-x)
x = (2+3y) / (1-y)
x - xy = 2 + 3y
x - 2 = xy + 3y = (x+3)y
y = (x-2) / (x+3)
h^(-1)(x) = (x-2) / (x+3)

g^(-1)(x) = h^(-1)(f(x)) = f^(-1){ (x-2)/(x+3) }

Answer 5

\[
(h \circ f)^{-1} = (f^{-1} \circ h^{-1})
\]

Answer 6

or
\[
(h(f(x)))^{-1} = f^{-1}(h^{-1}(x))
\]

Answer 7

Thank you !