Question

Let f(x)=e^1/2x, Find the function of fog (x), if gof(x) = x​

Answer 1

Since
`(g\circ f)(x) = g(f(x)) = x`, it follows that
`g` is the inverse of
`f`.

Then

`(g\circ f)(x) = \boxed{(f\circ g)(x) = x}`

Put another way, we have

`(g\circ f)(x) = g\left(e^(\frac12 x)\right) = x`

Apply the inverse of
`g` to both sides.

`g^(-1)\left(g\left(e^(\frac12 x)\right)\right) = e^(\frac12 x)= g^(-1)(x)`

but the left side is exactly
`f`, so
`f` is the inverse of
`g` and vice versa. Hence
`(f\circ g)(x) = (f\circ f^(-1))(x) = x`.