# Graphical description of $g(x)=f(1/x)$?

I am working on a problem set out of Spivak's Calculus, and I am stuck on the following problem:

Describe the graph of $$g(x)$$ in terms of the graph of $$f(x)$$. $$g(x)=f\left(\frac{1}{x}\right)$$

How can I possibly describe this when the function $$f(x)$$ is unknown? I've looked at several functions (e.g. $$\sin(x)$$, $$\ln(x)$$, $$x^n$$, …) but I don't see a clear pattern at all. Is there a certain property that $$g(x)$$ gives $$f$$? It seems that there is clearly not a single correct answer but I would like some sort of advice on how to construct a satisfactory response.

• $$\lim_{x\to0}g(x)=\lim_{x\to\infty}f(x)$$. In particular, if $$\lim_{x\to\infty}f(x)$$ doesn't exist, then $$\lim_{x\to0}g(x)$$ doesn't exist either.
• $$x_0$$ is a zero of $$g$$ if and only if $$\frac1{x_0}$$ is a zero of $$f$$. For instance, if $$f=\sin$$, the zeros of $$g$$ are the numbers of the form $$\frac1{n\pi}$$.
• $$g$$ is bounded if and only $$f$$ is bounded. Actually, they have the same $$\sup$$ and the same $$\inf$$.
• $$g$$ is increasing if $$f$$ is decreasing and $$g$$ is decreasing if $$f$$ is increasing.