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Let $f(x)=\dfrac{x}{x-7}$. Find a function $y=g(x)$ so that $(f\circ g)(x)=x$.

Does anyone have any ideas on how to do this? I've exhausted all sorts of possibilities and I am now clueless. (On a side note, this is not homework, but rather an optional online assignment I was given. It's really puzzling me.)

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Replace the $x$ with $y$ and the $f(x)$ with $x$, and solve for $y$. – J. M. Sep 10 '11 at 13:17
up vote 5 down vote accepted

So you want $$\frac{g(x)}{g(x)-7}= x \Longrightarrow g(x)= x\cdot g(x) - 7x $$

Solve for $g(x)$ from here.

If you still can't complete, then take a look below.

$$ x \cdot g(x)- g(x) = 7x \Longrightarrow g(x) \cdot \bigl(x-1\bigr) = 7x$$

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So is the solution 7x / x-1? – Mike Gates Sep 10 '11 at 13:28
@Mike: Yes, it is. – user9413 Sep 10 '11 at 13:31
@Mike: Mind your parentheses. :) Better to write it as 7x/(x-1) to avoid ambiguity. – J. M. Sep 10 '11 at 13:56

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