The "inspecting the graph" comment might refer to something like this.
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 $\frac{f(x) - g(x)}{g^{-1}(x) - f^{-1}(x)} \approx \frac{v(x) - w(x)}{-w(x) + v(x)} = 1$.
