Define f: R R by .f(x)= ( ⁻¹ ( ⁻¹ x ) ) ) , then _ x (f o f) x is equal to

Step-by-step solution

Given, f: R R by f(x)= [ ⁻¹ ( ⁻¹ x ) ] To find, _ x (f o f)(x) Let us find ( f f)(x) first, aligned & (f f)(x)=f(f(x)) & =f ( ( ⁻¹ ( ( ⁻¹ x ) ) ) . & =f ( ( ⁻¹ ( ( ⁻¹ x 1+x^2 ) ) ) ) & ⁻¹ x= ⁻¹ x 1+x^2 & =f ( ( ⁻¹ x 1+x^2 ) ) & ⁻¹ x= ⁻¹ 1 1+x^2 & =f ( ( ⁻¹ 1 1+ ( x 1+x^2 )^2 ) ) & aligned aligned & =f ( 1+x^2 1+2 x^2 ) (f f)(x) & = 1+ ( 1+x^2 1+2 x^2 )^2 1+2 ( 1+x^2 1+2 x^2 )^2 where, f(x)= 1+x^2 1+2 x^2 & = 1+2 x^2+1+x^2 1+2 x^2+2+2 x^2 = 2+3 x^2 3+4 x^2 aligned Let x=1 / y , then y 0 , when x gathered _ x 2+3 x^2