An inverse function can be found by switching the positions of the x and y variables of a function and (to put it in true function form) solving for y.
For example, if we have
f(x) = y = 3x -1
and we want to find the inverse of f, which we can call g, these are the steps to take:
1. Switch the positions of the x and y variables:
x = 3y - 1
2. Solve for y:
x + 1 = 3y
\frac{1}{3}x + \frac{1}{3} = y
So we have,
g(x) = \frac{1}{3}x + \frac{1}{3}
Now, watch what happens if we take f(g(x)):
f(g(x)) = f(\frac{1}{3}x + \frac{1}{3})
= 3(\frac{1}{3}x + \frac{1}{3}) - 1
= x + 1 - 1
= x
Similarly, if we try g(f(x)):
g(f(x)) = g(3x -1)
= \frac{1}{3}(3x -1) + \frac{1}{3}
= x - \frac{1}{3} + \frac{1}{3}
= x
This happens with any function and its true inverse. If we take the composite of a function and its inverse in either directions (i.e. f(g(x)) of g(f(x)) ), we end up getting x back. Or in the case of a number, e.g. f(g(1)) = g(f(1)) = 1
An inverse function basically "undoes" whatever a function does to a number.
So, your g(x) is not correct. And you may not always be able to easily find the inverse of a function solved for y, as is true with the function you have been given.
That being said, there are a couple ways to solve your problem.
The quickest way is to try playing around with the formula you gave under "relevant equations," to see if any expression simplifies, then take it from there.
Otherwise, you can look up a formula for the derivative of an inverse function to find g'(x) by skipping right to the derivative of an inverse function without finding the inverse. It can be found in any calculus textbook.
