The question comes from Calculus by Spivak - Ch 12 - 2) iv). It asks us to describe the graph of $f^{-1}$ when $f$ is decreasing and always negative. I interpret the inverse function as decreasing and not defining it for $x \geq 0$. But using the tools I have at my disposal, mainly the diagonal line test and drawings, I would get a function that is increasing but still defined only on for $x < 0$. This is also what the solution manual says: here is a screenshot of its solution.
I really don't agree with the drawing of the inverse function. I agree with how the original function, $f$, is drawn. Based on the tools this is correct, but it feels wrong......What may I be missing to reconcile this somewhat simple issue?
You are right, the graph of $f^{-1}$ is incorrect: I think that the graph of $f$ was rotated 90 degrees clockwise instead of being flipped with respect to the line $y=x$. It should diverge to $-\infty$ as $x\to \alpha^-=\lim_{x\to-\infty}f(x)$, and diverge to $\infty$ as $x\to\beta=\lim_{x\to\infty}f(x)$. Here, from the graph of $f$, it seems that $\alpha=0$ and $\beta=-\infty$, but this is not part of the hypothesis, strictly speaking.