Composition of functions

Question

Consider f(x)=(x+1)/(2x-4)

Find functions g and h such that f=g of h

Answer 1

So g(h(x)) must be (x+1)/(2x+4)

There are an unlimited amount of answers. I think it works best to just pick something for h(x) to start with. Picking h(x) = x seems way too trivial, so how about h(x) = 2x

Then you want to find some numerator for g(x) that multiplies with 2x to get "x+1" so how about 1/2x + 1

So now we have h(x) = 2x and we have the numerator of g(x) = 1/2x + 1 so now we just need a denominator for g(x) that with multiplies 2x to get 2x + 4 so that could be "x+4"

So, one answer could be h(x) = 2x and g(x) = (1/2x + 1)/(x + 4)

Another possibility is letting h(x) = 1/2x + 1, then g(x) would be (2x - 1)/(4x)