Let f(x) be a continuous function in [-1, 1] and satisfies `f(2x^2 - 1) = 2xf AA x in [-1, 1]`. Prove that f(x) is identically zero for all `x in [-1, 1]`

If a continuous function f on [0, a] satisfies f(x)f(a-x)=1, a >0, then find the value of int_0^a(dx)/(1+f(x))

Let f: R->R be a continuous function and f(x)=f(2x) is true AAx in R . If f(1)=3, then the value of int_(-1)^1f(f(x))dx is equal to

Let f: R rarr R be a continuous odd function, which vanishes exactly at one point and f(1)=1/2 . Suppose that F(x)=int_(-1)^xf(t)dt for all x in [-1,2] and G(x)=int_(-1)^x t|f(f(t))|dt for all x in [-1,2] . If lim_(x rarr 1)(F(x))/(G(x))=1/(14) , Then the value of f(1/2) is

Let f(x) be a continuous function defined for 1 le x le 3. If f(x) takes rational values for all x and f(2) = 10, then f(1,5) is equal to

Let f(x) be continuous functions f: RvecR satisfying f(0)=1a n df(2x)-f(x)=xdot Then the value of f(3) is 2 b. 3 c. 4 d. 5

Let f:RrarrR be a continuous odd function, which vanishes exactly at one point and f(1)=1/2. Suppose that F(x)=int_-1^xf(t)dt for all x in[-1,2] . If lim_(xto1)(F(x))/(G(x))=1/14 , then the value of f(1/2) is

A continuous real function f satisfies f(2x)=3f(x)AAx in RdotIfint_0^1f(x)dx=1, then find the value of int_1^2f(x)dx

Let f: RR to RR be a continuous odd function, which vanishes exactly at one point and f(1)=(1)/(2) . Suppose that F(x)=int(-1)^(x)f(t)dt for all x in[-1,2] and G(x)=int_(-1)^(x)t|f(ft)|dt" for all "x in[-1,2]." If "underset(xto1)lim(F(x))/(G(x))=(1)/(14) then the value of f((1)/(2)) is-

The function f:R->[-1/2,1/2] defined as f(x)=x/(1+x^2) is

Let f:[0,1] rarr [0,1] be a continuous function. Then prove that f(x)=x for at least one 0lt=xlt=1.