Question

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

Answer 1

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`