Ex. 1 Find all functions f : R + R given by f(x) = ax +...

Question

Question

Ex. 1 Find all functions f : R + R given by f(x) = ax + b, a, b € R, with the property that f (x + y) = f(x) + f(y) – 2a, Vw,

Answer 1

Answer #1

solution

we know that given f: R + Ris given by f (2) = ax +bER

and also given that the property f (x + y) = f (0) + f(y) - 2a Vr, YER

→ we can write that f () = ar + b from the given data a, b E IR

and f (x + y) = a (x + y) + b

similarly f (y) = ay + b

from that we can calculate

a(+y) + b = ax + b + ay +6 - 2a

common the similar terms we get

= a (x + y) + 26 – 2a

b=2b - 2a 2a = 2b-b 2a = b

sof(1) = ax + 2a = a (x+2) a EiR

Answer 2

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