about the function?

You must use brackets/parentheses for clear meaning !

f(x) = (x - 2)/(x + 2)

x = (y - 2)/(y + 2)

x(y + 2) = y - 2

solve for y...

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

I suspect that presentation is INCORRECT.
Could it be that what you ACTUALLY mean is :-
f (x) = (x - 2) / (x + 2)
Let y = f (x) = (x - 2) / (x + 2) = y and g (y) = x
g is inverse of f
x - 2 = y (x + 2 )
x - 2 = xy + 2y
(1 - y)x = 2y + 2
x = (2y + 2) / (1 - y)
g (y) = (2y + 2) / (1 - y)
g (x) = (2x + 2) / (1 - x)______inverse function.

To find the inverse function, first, I'll put this into y = form instead of f(x) = form:

f(x) = (x - 2) / (x + 2)
y = (x - 2) / (x + 2)

Now, switch the y for and x and all x's into y's, then solve for y:

x = (y - 2) / (y + 2)
x(y + 2) = y - 2
xy + 2x = y - 2
xy - y = -2x - 2
y(x - 1) = -2x - 2
y = (-2x - 2) / (x - 1)

I'm going to multiply both halves of that fraction by -1 to get the negatives out of the numerator:

y = (2x + 2) / (1 - x)

Now I'll change the y back into function form:

f⁻¹(x) = (2x + 2) / (1 - x)

x ≠ -2 and x ≠ 1

(-2 from what you gave me, then 1 from what I determined due to the new denominator)