If `f(x)=(h_1(x)-h_1(-x))(h_2(x)-h_2(-x))... (h_(2n+1)(x)-h_(2n+1)(-x)a n df(200)=0,` then prove that `f(x)` is many one function.

If f(x+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^2) is an even function.

If f(x+f(y))=f(x)+yAAx ,y in Ra n df(0)=1, then prove that int_0^2f(2-x)dx=2int_0^1f(x)dxdot

If g(x)=(lim)_(mvecoo)(x^mf(x)+h(x)+3)/(2x^m+4x+1) when x!=1a n dg(1)=e^3 such that f(x),g(x)a n dh(x) are continuous functions at x=1 then the value of 5f(1)-2h(1) is 7 b. 6 c. 9 d. 8

f(x)={(x-1",",-1 le x le 0),(x^(2)",",0le x le 1):} and g(x)=sinx Consider the functions h_(1)(x)=f(|g(x)|) and h_(2)(x)=|f(g(x))|. If for h_(1)(x) and h_(2)(x) are identical functions, then which of the following is not true?

If f(x)=2x+3, g(x) = 1-2x and h(x)=3x . Prove that f o (g o h) = (f o g) o h

If y=f(x)=((x+3))/((x-1)),t h e n x=f(y) (b) f(1)=3 y increases with xforx<1 f is a rational function of x

If f(x)=2x+3, g(x)=1-2x and h(x) =3x . Prove that fo(goh)=(fog)oh .

Let f(x)=(x^(5)-1)(x^(3)+1),g(x)=(x^(2)-1)(x^(2)-x+1) and let h(x) be such that f(x)=g(x)h(x) . Then lim_(xto1)h(x) is

Statement 1: If differentiable function f(x) satisfies the relation f(x)+f(x-2)=0AAx in R , and if ((d/(dx)f(x)))_(x=a)=b ,t h e n((d/(dx)f(x)))_(a+4000)=bdot Statement 2: f(x) is a periodic function with period 4.

If y=f(x)=((x+2))/((x-1)),t h e n (a) x=f(y) (b) f(1)=3 (c) y increases with x for x<1 (d) f is a rational function of x