Project ForgeDiscrete Mathematics
Set Theory & Abstract Algebra
GATE 2015
Hard
If g(x) = 1 − x and h(x) = x/(x−1), then g(h(x))/h(g(x)) is:
- h(x)/g(x)
- B−1/x
- Cg(x)/h(x)
- Dx/(1−x)²
Explanation
g(h(x)) = 1 − x/(x−1) = (x−1−x)/(x−1) = −1/(x−1). h(g(x)) = g(x)/(g(x)−1) = (1−x)/(1−x−1) = (1−x)/(−x) = (x−1)/x. So g(h(x))/h(g(x)) = [−1/(x−1)] / [(x−1)/x] = −x/(x−1)². Meanwhile h(x)/g(x) = [x/(x−1)]/(1−x) = x/[(x−1)(1−x)] = x/[−(x−1)²] = −x/(x−1)². These match exactly, so the answer is h(x)/g(x).
Source: GATE CS 2015 (Session 1 / 7th Feb Shift 1) (2015)
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