For x in mathbb R - 0,1 , let f_1(x) = frac 1 x , f_2(x) = 1-x and f_3(x) = frac 1 1-x be three given functions. If a function J(x) satisfies (f_2 circ J circ f_1)(x) = f_3(x), then J(x) is equal to:

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EEJ MAIN Mathematics QUESTION #7482

Question 1

For \(x \in \mathbb{R} - \{0,1\}\), let \(f_1(x) = \frac{1}{x}\), \(f_2(x) = 1-x\) and \(f_3(x) = \frac{1}{1-x}\) be three given functions. If a function J(x) satisfies \((f_2 \circ J \circ f_1)(x) = f_3(x)\), then J(x) is equal to:

\(f_3(x)\)✔️
\(\frac{1}{x}f_3(x)\)
\(f_2(x)\)
\(f_1(x)\)

Correct Answer Explanation

We have \((f_2 \circ J \circ f_1)(x) = f_2(J(f_1(x))) = f_2(J(\frac{1}{x})) = 1 - J(\frac{1}{x}) = f_3(x) = \frac{1}{1-x}\). So \(J(\frac{1}{x}) = 1 - \frac{1}{1-x} = \frac{1-x-1}{1-x} = \frac{-x}{1-x} = \frac{x}{x-1}\). Replacing \(\frac{1}{x}\) with t (so \(x = \frac{1}{t}\)): \(J(t) = \frac{1/t}{1/t - 1} = \frac{1}{1-t} = f_3(t)\).

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