If g(x) = 1 - x and h(x) = x/(x-1), then g(h(x))/h(g(x)) is: A. h(x)/g(x) B. -1/x C. g(x)/h(x) D. x/(1-x)^2

GATE 2015 · Discrete Mathematics · Functions · medium

Answer: A. h(x)/g(x)

1. Compute g(h(x)): g(h(x)) = 1 - x/(x-1) = ((x-1) - x)/(x-1) = -1/(x-1).
2. Compute h(g(x)): h(g(x)) = (1-x)/((1-x) - 1) = (1-x)/(-x) = (x-1)/x.
3. Form and simplify the ratio: = -1/(x-1) * x/(x-1) = -x/(x-1)^2.
4. Match to an option: h(x)/g(x) = x/((x-1)(1-x)) = x/(-(x-1)^2) = -x/(x-1)^2, which equals the ratio. So the answer is option A.