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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.

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A few suggestions:

  • $\lim_{x\to0}g(x)=\lim_{x\to\infty}f(x)$. 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.
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