For x in R - {0, 1},let {f_1}(x) = frac{1}{x} | vedclass

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(A) Given functions are ${f_1}(x) = \frac{1}{x}$,${f_2}(x) = 1 - x$,and ${f_3}(x) = \frac{1}{1 - x}$.The given equation is $(f_2 \circ J \circ f_1)(x)

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${f_3}(x)$

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(A) Given functions are ${f_1}(x) = \frac{1}{x}$,${f_2}(x) = 1 - x$,and ${f_3}(x) = \frac{1}{1 - x}$.The given equation is $(f_2 \circ J \circ f_1)(x) = f_3(x)$,which can be written as ${f_2}(J(f_1(x))) = f_3(x)$.Substituting the expressions for ${f_2}$ and ${f_3}$:$1 - J(f_1(x)) = \frac{1}{1 - x}$.Rearranging to solve for $J(f_1(x))$:$J(f_1(x)) = 1 - \frac{1}{1 - x} = \frac{1 - x - 1}{1 - x} = \frac{-x}{1 - x} = \frac{x}{x - 1}$.Since ${f_1}(x) = \frac{1}{x}$,let $t = \frac{1}{x}$,which implies $x = \frac{1}{t}$.Substituting $x = \frac{1}{t}$ into the expression for $J(f_1(x))$:$J(t) = \frac{\frac{1}{t}}{\frac{1}{t} - 1} = \frac{\frac{1}{t}}{\frac{1 - t}{t}} = \frac{1}{1 - t}$.Thus,$J(x) = \frac{1}{1 - x}$,which is equal to ${f_3}(x)$.

For $x \in R - \{0, 1\}$,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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