Let `A=(x in R: x ge 1)`. The inverse of the function of `f:A to A` given by `f(x)=2^(x^((x-1))`. IsA. `((1)/(2))^(x^((x-1)))`B. `(1)/(2){1+sqrt(1+4log_(2)x)}`C. `(1)/(2){1-sqrt(1+4log_(2)x)}`D. None of these

Let `A=(x in R: x ge 1)`. The inverse of the function of `f:A to A` gi

1.	Let `A=(x in R: x ge 1)`. The inverse of the function of `f:A to A` given by `f(x)=2^(x^((x-1))`. IsA. `((1)/(2))^(x^((x-1)))`B. `(1)/(2){1+sqrt(1+4log_(2)x)}`C. `(1)/(2){1-sqrt(1+4log_(2)x)}`D. None of these
Answer» Correct Answer - B It can be easily verified that `f:A to A` is a bijection. `"Let"f(x)=y"Then"` `f(x)=y` `Rightarrow 2^(x^((x-1)))=y``Rightarrow x(x-1)=log_(2)y` `Rightarrow x^(2)-x-log_(2) y=0` `x=(1pmsqrt(1+4log_(2)y))/(2)` `Rightarrow x=(1)/(2) [1+sqrt(1+4log_(2)y)]" "[therefore x lt 1]` `Rightarrow f^(-1)(y)=(1)/(2){1+sqrt(1+4log_(2)y)}` `Rightarrow f^(-1)(y)=(1)/(2){1+sqrt(1+4log_(2)y)}` x`f^(-1)(x)=(1)/(2) {1+sqrt(1+4log_(2) x)}`