If `f(x)=(a-x^(n))^(1/n),agt0` and n is a positive integer, then prove that f(f(x)) = x.

If f(x)=(a-x^n)^(1/n)," where "a gt0 and n is a positive integer, show that f[f(x)]=x.

If f(x) = (p - x^n)^(1/n), p gt 0 and n is positive integer, then the value of f[f(x)]

If f(x)=(1-x)/(1+x) , then prove that f(x)+f(1/x)=0 .

If f(x)=x^3+b x^2+c x+d and f(0),f(-1) are odd integers, prove that f(x)=0 cannot have all integral roots.

If f(x)=x^n,n being a non-negative integer, then the values of n for which f'(alpha+beta)=f'(alpha)+f'(beta) for all alpha,beta>0 is

If (2+sqrt(3))^n=I+f, where I and n are positive integers and 0lt f lt1 , show that I is an odd integer and (1-f)(1+f) =1

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.

For any positive integer n , define f_n :(0,oo)rarrR as f_n(x)=sum_(j=1)^ntan^(-1)(1/(1+(x+j)(x+j-1))) for all x in (0, oo) . Here, the inverse trigonometric function tan^(-1)x assumes values in (-pi/2,pi/2)dot Then, which of the following statement(s) is (are) TRUE? sum_(j=1)^5tan^2(f_j(0))=55 (b) sum_(j=1)^(10)(1+fj '(0))sec^2(f_j(0))=10 (c) For any fixed positive integer n , (lim)_(xrarroo)tan(f_n(x))=1/n (d) For any fixed positive integer n , (lim)_(xrarroo)sec^2(f_n(x))=1

Let f(x)a n dg(x) be two function having finite nonzero third-order derivatives f'''(x) and g'''(x) for all x in R . If f(x).g(x)=1 for all x in R , then prove that (f''')/(f') - (g''')/(g') = 3((f'')/f - (g'')/g) .

Prove that (""^n C_0)/x-(""^n C_1)/(x+1)+(""^n C_2)/(x+2)-.....+(-1)^n(""^n C_n)/(x+n)=(n !)/(x(x+1) . . . (x-n)), where n is any positive integer and x is not a negative integer.