Use inverse functions to determine these are Inverses f(x)=2x–2/x+3 & g(x)=3x+2/2–x

Checking my work is correct. Thank you

f(x) = (2x - 2)/(x + 3)
Let y = f(x)
=> y = (2x - 2)/(x + 3)
=> xy + 3y = 2x - 2
=> xy - 2x = -3y - 2
=> x = (3y + 2)/(2 - y)
replacing y with x
we get
inverse of f(x) = (3x + 2)/(2 - x) = g(x)

Also f(g(x)) = (2(3x + 2)/(2 - x) - 2)/((3x + 2)/(2 - x) + 3)
= (6x + 4 - 4 + 2x)(2 - x)/[(2 - x)(3x + 2 + 6 - 3x)
= (8x)(2 - x)/[(2 - x)(8)]
= x
And
g(f(x))
= (3(2x - 2)/(x + 3) + 2)/(2 - (2x - 2)/(x + 3))
= (6x - 6 + 2x + 6)/(2x + 6 - 2x + 2)
= 8x/8
= x

Have first of all to check if questions are correct.

I would imagine that what you mean is in fact :-

f (x) = (2x - 2) / (x + 3) and g(x) = (3x + 2) / (2 - x)

and NOT as given.

Why do you need to check any work. Determining if two functions are inverses of each other is a very straightforward procedure you cannot possibly get it wrong.