If g(x) = 1 - x and h(x) = xx-1, then g(h(x))h(g(x)) is:

GATE 2015 CS – Question 11

Engineering Mathematics · Discrete Mathematics: Sets, Relations, Functions, Partial Orders and Lattices · 1 mark · Multiple choice

If $g(x) = 1 - x$ and $h(x) = \frac{x}{x-1}$, then $\frac{g(h(x))}{h(g(x))}$ is:

$\frac{h(x)}{g(x)}$
$\frac{-1}{x}$
$\frac{g(x)}{h(x)}$
$\frac{x}{(1-x)^2}$

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Correct answer: (A) $\frac{h(x)}{g(x)}$

Explanation

$g(h(x)) = 1 - \frac{x}{x-1} = \frac{-1}{x-1}$ and $h(g(x)) = \frac{1-x}{(1-x)-1} = \frac{x-1}{x}$. Their ratio is $\frac{-1}{x-1} \cdot \frac{x}{x-1} = \frac{-x}{(x-1)^2}$. Also $\frac{h(x)}{g(x)} = \frac{x}{(x-1)(1-x)} = \frac{-x}{(x-1)^2}$, which matches.