A function describes $g(x + y) = g(x)g(y)$ for all $x, y$. If $g(4) = + 3,$ find the value of $g(–8)$?

Question

I tried solving the question, but I always ended up getting my answer wrong. I'm also not sure if the given options are correct.

Here are the options that were given:

A. 1/3

B. 1/9

C. 9

D. 6

Answer 1

$$g(4)=g(0+4)=g(0)g(4)\implies g(0)=1$$

so $$1=g(4-4) = g(4)g(-4)\implies g(-4) = {1\over 3}$$

so $$g(-8) = g(-4+(-4))= g(-4)^2 ={1\over 9}$$

Answer 2

What is $g(0)$? What is $g(-4)$, knowing $g(0)$ and $g(4)$? What is $g(-8)$, knowing $g(-4)$?

Answer 3

$g(4) = g(2)^2 \implies g(2) = \sqrt{3}$

$g(4-2) = g(4)g(-2) \implies g(-2) = \frac{1}{\sqrt{3}}$

$g(-2-2) = g(-4) = g(-2)^2 = \frac{1}{3}$

$g(-4-4) = g(-8) = g(-4)^2 = \frac{1}{9}$