3.If `f(x)=x^(2)-(1)/(x^(2))`,then prove that: `f(x)+f((1)/(x))=0`

(i) If f(x) = x + (1)/(x) prove that: [f(x)]^(3) = f(x^(3)) + 3f ((1)/(x)) (ii) If f(x) = x^(3) - (1)/(x^(3)) , prove that f(x) + f((1)/(x)) = 0 (iii) If f(x) = (1-x^(2))/(1+ x^(2)) , prove that f (tan theta) = cos 2 theta

If f(x)=(x^(2)-1)/(x^(2)+1) prove that f((1)/(x))=-f(x)

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

If f(x)=(x^(2))/(1+x^(2)), prove that lim_(x rarr oo)f(x)=1

If f(x)=x+(1)/(x) , then prove that : {f(x)}^(3)=f(x^(3))+3*f((1)/(x))

For a function f(x),f(0)=0 and f'(x)=(1)/(1+x^(2)), prove that f(x)+f(y)=f((x+y)/(1-xy))

If f(x)=log[(1+x)/(1-x)], then prove that f[(2x)/(1+x^(2))]=2f(x)

If f(x)="log"(1+x)/(1-x) , then prove that: f((2x)/(1+x^(2)))=2f(x)

If f(x)=(a-x^(n))^((1)/(n)),a>0 and n in N, then prove that f(f(x))=x for all x.

If f(x) is monotonically increasing function for all x in R, such that f''(x)gt0andf^(-1)(x) exists, then prove that (f^(-1)(x_(1))+f^(-1)(x_(2))+f^(-1)(x_(3)))/(3)lt((f^(-1)(x_(1)+x_(2)+x_3))/(3))