The "inspecting the graph" comment might refer to something like this.

Consider two smooth curves $y = f(x)$ and $y = g(x)$ that are tangent to $y = x$ at $(0,0)$. For $x$ near 0, define $u(x)$ and $v(x)$ so that $g^{-1}(x) - f^{-1}(x) = u(x)$ and $f(x) - g(x) = v(x)$. In the picture, $v(x) = BC$ and $u(x) = AD$. But both curves have slope very close to 1, so $AD \approx BC$, i.e. $u(x) \approx - v(x)$, and $\frac{f(x) - g(x)}{g^{-1}(x) - f^{-1}(x)} \approx 1$.

alt text http://www.math.ubc.ca/%7Eisrael/problems/barrow.jpg

The "inspecting the graph" comment might refer to something like this.

Consider two smooth curves $y = f(x)$ and $y = g(x)$ that are tangent to $y = x$ at $(0,0)$. For $x$ near 0, define $u(x)$ and $v(x)$ so that $g^{-1}(x) - f^{-1}(x) = u(x)$ and $f(x) - g(x) = v(x)$. In the picture, $v(x) = BC$ and $u(x) = AD$. But both curves have slope very close to 1, so $AD \approx BC$, i.e. $u(x) \approx - v(x)$, and $\frac{f(x) - g(x)}{g^{-1}(x) - f^{-1}(x)} \approx 1$.

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 $\frac{f(x) - g(x)}{g^{-1}(x) - f^{-1}(x)} \approx \frac{v(x) - w(x)}{-w(x) + v(x)} = 1$

alt text http://www.math.ubc.ca/%7Eisrael/problems/barrow.jpg

The "inspecting the graph" comment might refer to something like this.

Consider two smooth curves $y = f(x)$ and $y = g(x)$ that are tangent to $y = x$ at $(0,0)$. For $x$ near 0, define $u(x)$ and $v(x)$ so that $g^{-1}(x) - f^{-1}(x) = u(x)$ and $f(x) - g(x) = v(x)$. In the picture, $v(x) = BC$ and $u(x) = AD$. But both curves have slope very close to 1, so $AD \approx BC$, i.e. $u(x) \approx - v(x)$, and $\frac{f(x) - g(x)}{g^{-1}(x) - f^{-1}(x)} \approx 1$.

alt text http://www.math.ubc.ca/%7Eisrael/problems/barrow.jpg