Cosider the functions f and g defined as follows.
f(x) = x-3/x-2 for all x E R, does not equal 2.
g(x) = 3-x for all x E R.
Find all possible functions that are generated by composing the functions f and g. After finding all functions, prove that you have them all and identify the domain of each. You will want to look at (f of g), g of f, f of f, g of g, f of(g of f) and so on.

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Composing functions just means you sub one function into the other.

For instance, f of g. Plug g in for x in f.

f(g(x))=\frac{(3-x)-3}{(3-x)-2}=\frac{-x}{1-x}=\frac{x}{x-1}

f(f(x))=\frac{\frac{x-3}{x-2}-3}{\frac{x-3}{x-2}-2}=\frac{2x-3}{x-1}

Now, you try the others and let's see what you get.

The domain of the first and last ones is everything except 1, because x=1 will result in division by 0.

Is this even logical?

What do you mean, is it logical? They are just composite functions.