If \(g(x) = 1 - x\) and \(h(x) = \frac{x}{x-1}\) , then…
2015
If \(g(x) = 1 - x\) and \(h(x) = \frac{x}{x-1}\) , then \(\frac{g(h(x))}{h(g(x))}\) is:
- A.
\(\frac{h(x)}{g(x)}\) - B.
\(\frac{-1}{x}\) - C.
\(\frac{g(x)}{h(x)}\) - D.
\(\frac{x}{(1-x)^{2}}\)
Attempted by 153 students.
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Correct answer: A
Solution:
Compute g(h(x)): g(h(x)) = 1 - h(x) = 1 - x/(x-1) = ((x-1)-x)/(x-1) = -1/(x-1).
Compute h(g(x)): h(g(x)) = h(1-x) = (1-x)/((1-x)-1) = (1-x)/(-x) = (x-1)/x.
Form the quotient: g(h(x))/h(g(x)) = [-1/(x-1)] / [(x-1)/x] = -1/(x-1) * x/(x-1) = -x/(x-1)^2.
Show equality with h(x)/g(x): h(x)/g(x) = [x/(x-1)]/(1-x) = x/[(x-1)(1-x)] = x/[-(x-1)^2] = -x/(x-1)^2, so it matches the quotient computed above.
Note the domain restrictions: x ≠ 1 (from h) and x ≠ 0 (from h(g(x))).
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