Pure Mathematics 1 by Backhouse

Ex:2f q13

The real function f, defined for all $\displaystyle x \epsilon \mathbb{R} $, is said to be multiplicative if, for all $\displaystyle y \epsilon \mathbb{R} $, $\displaystyle x \epsilon \mathbb{R} $,

f(xy) = f(x)f(y)

Q: Prove that if f is multiplicative function then

a) either f(0) =0 or f(x)=1

b) either f(1) = 1 or f(x) = 0

c) $\displaystyle f(x)^{n} $ = $\displaystyle \left \{ f(x) \right \}^{n} $ for all positive integers n.

Give example of a non- constant multiplicative function

My attempt:

I have tried to apply what I have learn from the chapter of functions from this book but there is no bit on multiplicative functions, I have only learnt about odd & even functions. I tried but failed please help

let f(x) = even functions thus f(a) = f(-a)

f(y) = odd functions thus - f(b) = f(-b)

f(ab) = f(a)f(b)

= f(-a) (-f(b)) = -f(-a)f(-b) = -f(a)f(-b)

hence i am stuck please help. I have tried to look at youtube and search for multiplicative functions but it seems to be advanced number theory. I dont have knowledge in that area thus please can an expert give an answer that does not have advance mathematical symbols as my maths level is at high school

Thank you very much for your help in advance