Let the domain of the function f be all the real numbers. It is known that for all x in this domain, f(x+1)=2 f(x) . Also, for x (0,1], f(x)=x^2-x . If f(x) -8 9 for x (- , m] . Find 3 m .

Step-by-step solution

aligned & f(x+1)=2 f(x) x R ...(1) & f(x)=x^2-x x (0,1] ...(2) & f(x) -8 9 (=-0.8) x (- , m) ...(3) & 3 m=7 & x x+1 in Eqn. (1) & f(x+2)=2 f(x+1) 2^2 f(x) & f(x+3)=2^n f(x)=2^n (x^2-x ) & x+n=t & f(t)=2^n [(t-n)^2-(t-n) ] where t (n, n+1] & f(t)=2 n [t^2-t(2 n+1)+n^2+n ] aligned array lr 2^n [ (t- 2 n+1 2 )^2- 1 4 ] & f(t) -2^ n-2 array Where array ll n=1 f(t) - 1 2 & t (1,2] n=2 f(t) -1 & t (1,2] array -0.8 will come in this region we will get two values of t and we will take smaller value of t as x (- , m) . alig