I am working on composite functions, but I've gotten to a point that stumps me; I don't have any of this in my reading. I have f(x)= x/2 g(x)= 2/x...

Hello!

The way you use it, (-1) becomes a factor (you multiply the left part of your expression by this (-1)). But it is incorrect, (-1) isn't a factor, it is an argument of the composite function f/g. Actually all your steps are correct except the last one, where you should substitute x=-1, not multiply by -1.

I want to explain it from the beginning: if we have two functions, f(x) and g(x), then we may create a function h=(f/g) by the rule

(for those x's where of course).

Then we can substitute x with any specific value, for example -1. In our case, (f/g)(x) is equal to

For this is equal to 1/4 (this is the answer). Note that we obtain a specific number, not an expression involving x.

Also, we can substitute x=-1 before simplifying,

The answer is of course the same.