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Question
Let the function f satisfy f(xy) = f(x)/y for all positive numbers x,y. If f(5) = 10 then what is the value of f(8). Thank you.

In order to figure this out, I treated y like an arbitrary scaling factor, α, which scales the argument x in direct proportion and the output f(x) in  inverse proportion. This is reminiscent of a hyperbola.

I rearranged the expression so that

α = f(x)/f(αx) = "constant"

the word "constant" being in parentheses because the expression should hold for any α (which is really y, for now). The expression also holds for all x, in particular x = 1, so with α = 5 we have f(α) = f(5) = 10 and

f(1) = αf(α) =5f(5) = 5･10 = 50.

Taking f(x) to be a hyperbola, we have f(x) = c/x, where c is a constant given by f(1) = c = 50.

Thus, f(8) = 50/8 = 6.25.
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#### randy patton

##### Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

##### Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

Publications
J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

Education/Credentials
M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

Past/Present Clients
Also an Expert in Oceanography