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

Question

asked Sep 14, 2023 6.9k views

1 Answer

Milliron X · Answer 1 · 2023-09-19T16:40:44+0000

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

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
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
, so
is the inverse of
and vice versa. Hence
$(f\circ g)(x) = (f\circ f^(-1))(x) = x$ .