Prove or disprove ? f(A/B)=f(A)/f(B)

1 Answer
Jun 12, 2017

This identity is generally false...

Explanation:

In general this will be false.

A simple example would be:

f(x) = 2

Then:

f(1/1) = 2 != 1 = 2/2 = f(1)/f(1)

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Bonus

For what kind of functions f(x) does the identity hold?

Note that:

f(1) = f(1/1) = f(1)/f(1) = 1

f(0) = f(0/x) = f(0)/f(x)" " for any x

So either f(0) = 0 or f(x) = 1 for all x

If n is any integer and:

f(x) = x^n

Then:

f(a/b) = (a/b)^n = a^n/b^n = f(a)/f(b)

There are other possibilities for f(x):

f(x) = abs(x)^c" " for any real constant c

f(x) = "sgn"(x)*abs(x)^c" " for any real constant c