Why isn't f(x^2) a horizontal stretch of f(x) by a factor of "1/x"?

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I know this question seems silly, but it came to mind while reading about transforming functions. Is the statement "y=f(kx) results from scaling the graph of y=f(x) horizontally by a factor of 1/k" not applicable when k=x? A thorough explanation would be appreciated.

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I would consider for $x$ positive that $f(x^2)$ is simply a variable horizontal stretch. That is for $0<x<1$, you get a horizontal 'widening' and for $x>1$ you get a horizontal 'narrowing'. And (see my comment above) you have to remember that for $x$ negative, you have to take the positive $x-$axis values and reflect them over the $y-$axis. So for example this is how you would apply your rule to $f(x)=x$.

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In the context discussed in the book, it is assumed that $k$ is a constant. So setting $k=x$ is not allowed as we are considering $x$ to be a variable.