# f(x+y)=f(x)+f(y) , f(1)=3 , f(x)=? pls explain !!

Rico Grant | Certified Educator

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We are given that f(x+y)=f(x)+f(y) and that f(1)=3.

We assume that f(x) is continuous. (A non-continuous image is possible, but requires real analysis which is beyond a high school class.)

Then f is of the form f(x)=kx (or y=kx) for some `kinRR` .(f(0+0)=f(0)+f(0)==>f(0)=2f(0) so f(0)=0. Also f(rx)=f(x)+f(x)+ ... +f(x) (r times)==>f(rx)=rf(x).

So plugging in the known values we get `3=k*1...

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