Altering the curvature of an asymptotic curve?

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With a basic function like $y = 1/x$ mapped over the first 100 integers I see the curve elbow at around 5.

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I want to shift this closer to 100... to "flatten" the curve so to speak, so it grows more quickly on the right of the elbow and less so on the left.

How can I accomplish this?

I've tried things like $y = 1/(x^C)$ where $C$ is some suitable value $< 1$ but all it does is shift the right end of the curve with the elbow at the same place