A continuous function `f(x)onRvec`satisfies the relation `f(x)+f(2x+y)+5x y=f(3x-y)+2x^2+1forAA x ,y in RdotT h e n findf(x)dot`

A continuous function f(x) satisfies the relation f(x)=e^x+int_0^1 e^xf(t)dt then f(1)=

A function f: R -> R satisfy the equation f (x)f(y) - f (xy)= x+y for all x, y in R and f(y) > 0 , then

Suppose the function f(x) satisfies the relation f(x+y^3)=f(x)+f(y^3)dotAAx ,y in R and is differentiable for all xdot Statement 1: If f^(prime)(2)=a ,t h e nf^(prime)(-2)=a Statement 2: f(x) is an odd function.

An even polynomial function f(x) satisfies a relation f(2x)(1-f(1/(2x)))+f(16x^2y)=f(-2)-f(4x y)AAx ,y in R-{0}a n df(4)=-255 ,f(0)=1. Then the value of |(f(2)+1)//2| is_________

If a function 'f' satisfies the relation f(x)f^('')(x)-f(x)f^(')(x) -f^(')(x)^(2)=0 AA x in R and f(0)=1=f^(')(0) . Then find f(x) .

Determine the function satisfying f^2(x+y)=f^2(x)+f^2(y)AAx ,y in Rdot

A function f: RvecR satisfies the equation f(x)f(y)-f(x y)=x+yAAx ,y in Ra n df(1)>0 , then f(x)f^(-1)(x)=x^2-4 b. f(x)f^(-1)(x)=x^2-6 c. f(x)f^(-1)(x)=x^2-1 d. none of these

If the function / satisfies the relation f(x+y)+f(x-y)=2f(x),f(y)AAx , y in R and f(0)!=0 , then (a) f(x) is an even function (b) f(x) is an odd function (c) If f(2)=a ,then f(-2)=a (d) If f(4)=b ,then f(-4)=-b

If f(x) satisfies the relation f(x+y)=f(x)+f(y) for all x , y in Ra n df(1)=5,t h e n (a) f(x)i sa nod dfu n c t ion (b) f(x) is an even function (c) sum_(n=1)^mf(r)=5^(m+1)C_2 (d) sum_(n=1)^mf(r)=(5m(m+2))/3

A real valued function f (x) satisfies the functional equation f (x-y)= f(x) f(y) - f(a-x) f(a +y) where a is a given constant and f (0) =1, f(2a -x) is equal to