For the function $$ f $$, use composition of functions to show that $$ f^{-1} $$ is as given. $$ f(x)=\frac{3-x}{x}, f^{-1}(x)=\frac{3}{x+1} $$ To show that $$ f^{-1}(x)=\frac{3}{x+1} $$, find $$ \left(f^{-1} \circ f\right)(x) $$ and $$ \left(f \circ f^{-1}\right)(x) $$ and check to see that each is $$ x $$. Beginning with $$ \left(f^{-1} \circ f\right)(x) $$. $$ \left(f^{-1} \circ f\right)(x)=\square $$

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Bush Burton · Accepted Answer

To show that $$ f^{-1}(x) = \frac{3}{x+1} $$ is the inverse of $$ f(x) = \frac{3-x}{x} $$, we compute $$ (f^{-1} \circ f)(x) $$ and $$ (f \circ f^{-1})(x) $$. Both computations yield $$ x $$, confirming that $$ f^{-1}(x) $$ is the inverse of $$ f(x) $$.

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