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Consider the two smooth curve curves $y = f(x)$ and $y = g(x)$ that is are tangent to $y = x$ at $(0,0)$. For $x$ near 0, define $u(x)$ and $v(x)$ so that $f(x g^{-1}(x) - u(x)f^{-1}(x) = x$ u(x)$and$f(x) - g(x) = x + v(x)$. In the picture, where$u$v(x) = BC$ and $v$ are small and positiveu(x) = AD$. Since the But both curves have slope of the graph from$(x - u, x)$very close to$(x,x+v)$is approximately 1, so$u AD \approx v$BC$, i.e. $f^{-1}(x) \approx x - u(x)$. Similarly, if $y = g(x) = x + w(x)$ is another such curve, $g^{-1}(x) u(x) \approx x - w(x)$v(x)$, and so$\frac{f(x) - g(x)}{g^{-1}(x) - f^{-1}(x)} \approx \frac{v(x) - w(x)}{-w(x) + v(x)} = 1$. 1 The "inspecting the graph" comment might refer to something like this. Consider the smooth curve$y = f(x)$that is tangent to$y = x$at$(0,0)$. For$x$near 0, define$u(x)$and$v(x)$so that$f(x - u(x)) = x$and$f(x) = x + v(x)$, where$u$and$v$are small and positive. Since the slope of the graph from$(x - u, x)$to$(x,x+v)$is approximately 1,$u \approx v$, i.e.$f^{-1}(x) \approx x - u(x)$. Similarly, if$y = g(x) = x + w(x)$is another such curve,$g^{-1}(x) \approx x - w(x)$, and so$\frac{f(x) - g(x)}{g^{-1}(x) - f^{-1}(x)} \approx \frac{v(x) - w(x)}{-w(x) + v(x)} = 1\$