Question

Need a hand proving that - f(xy) does not equal f(x) + f(y)

Answer 1

My suggestion is to find a counterexample. That is, pick a couple of simple functions and show that it's not true for those functions.

Answer 2

f(x)=2 , f(y)=3
f(xy)=2*3=6
f(x)+f(y)=2+3=5

Answer 3

^ That is bad example. Please don't write that.

Answer 4

Definition of the function is needed at first.

Answer 5

Finding one counter example should do, but please operate with functions that actually have a variable inside of them.

An example of this is:

f(x)= x+1.

f(xy) = xy +1
f(x) + f(y) = x+1+y+1=x+y+2

Asuming they are equal we have that:

xy+1=x+y+2 or xy=x+y+1.

Let x=1 and y=0, we have that 0=1 which is false and therefore:

f(x)+f(y) does not equal f(xy) for any value of x and y and any form of f(x).

Note that it does not mean that there are no functions and/or specific values so that f(x)+f(y)=f(xy), just that it is not something universally applicable to any imaginable function.

This is like saying that every bird in Europe flies.
If I show you one bird from Europe that doesn't fly it means that your statement is false however that doesn't mean that

- every bird can't fly
- birds in Europe can't fly

but it still does negate the original statement, that every bird in Europe flies.