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The book "A student's Guide to Fourier Transforms" 3rd edition, by J.F.James, published by C.U.P., on pages 20 and 21 quotes the equation $y = (x-1)\sqrt{x}$ as an example of one that does not meet the criteria for a Fourier transform because it is a two valued function. However the graph given is triple valued in places, with a continuous loop like that traced by an aeroplane looping the loop.

I cannot reconcile these, so please can anybody help with making sense of it, perhaps by showing what the plot of this equation really does look like?

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The footnote or figure in the book is incorrect. The equation $(1 - y) (x^2 + y^2) = 2 x^2$ describes a graph similar to the figure in the book.

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$y=(x-1)\sqrt x$ is nicely single valued if you view $y$ as a function of $x$. It is only defined for $x \ge 0$. It does have a vertical tangent at $x=0$. If you view $x$ as a function of $y$, it is double valued on $y \le 0, x \le 1$ A plot is below

enter image description here

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