Let f: R R be a function such that f(x+y)=f(x)+f(y) for all x, y R , and g: R (0, ) be a function such that g(x+y)=g(x) g(y) for all x, y R . If f ( -3 5 )=12 and g ( -1 3 )=2 , then the value of (f ( 1 4 )+g(-2)-8 ) g(0) is . ________.

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f(x+y)=f(x)+f(y) ...(1) f ( nx )= nf ( x ) n N ...(2) Now put y=-x in eq.(1) aligned & f ( x )+ f (- x )= f (0) f (0)=0 & f (- x )=- f ( x ) aligned f is odd function from eq. (2) from eq. (2) aligned & f(-n x)= nf (-x) & f (- nx )=-n f( x ) & aligned f ( mx )= mf ( x ) m Z ...(3) from eq. (2) and eq. (3) f ( nx )= nf ( x ) n Z ...(4) Now put x= p q where p , q Z , q 0 f ( np q )= nf ( p q ) n Z put n = q f(p)=q f ( p q ) pf (1)= qf ( p q ) from eq.(4) Let f (1)= a then pa = qf ( p q ) f ( p q )= ap q f ( x )= ax x

Let f: R R be a function such that f(x+y)=f(x)+f(y) for all x, y R , and g: R (0, ) be a function such that g(x+y)=g(x) g(y) for all x, y R . If f ( -3 5 )=12 and g ( -1 3 )=2 , then the value of (f ( 1 4 )+g(-2)-8 ) g(0) is . ________.

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