f-R-R-g-R-R-f-x-y-f-x-g-y-g-xy-f-x-g-y-f-x-g-x-

Question Number 1125 by 123456 last updated on 18/Jun/15

f : R \to R

g : R \to R

f (x + y) = f (x) + g (y)

g (x y) = f (x) g (y)

f (x) = ?

g (x) = ?

Commented by 123456 last updated on 18/Jun/15

f (0) = f (0) + g (0) \Rightarrow g (0) = 0

f (x) = f (x) + g (0) \Rightarrow g (0) = 0

g (0) = f (0) g (0) \Rightarrow f (0) = 1 \lor g (0) = 0

g (0) = f (x) g (0) \Rightarrow f (x) = 1 \lor g (0) = 0

g (1) = f (1) g (1) \Rightarrow f (1) = 1 \lor g (1) = 0

g (x) = f (1) g (x) \Rightarrow f (1) = 1 \lor g (x) = 0

g (- 1) = f (1) g (- 1) \Rightarrow f (1) = 1 \lor g (- 1) = 0

Answered by prakash jain last updated on 20/Jun/15

g (x) = f (x) g (1)

g (1) = 1 / k

f (x) = k g (x)

g (x y) = k g (x) g (y)

g (x) = x, k = 1

f (x) = k g (x) = x

Commented by prakash jain last updated on 20/Jun/15

g (x y) = \frac{g (x) g (y)}{g (1)} \Rightarrow g (x) = x^{r}, g (1) = 1

f (x) = x^{r}

f (x + y) = f (x) + g (y)

{(x + y)}^{r} = x^{r} + y^{r} \Rightarrow r = 1

g (x) = f (x) = x