Answer to Question #115469 in Algebra for roopak

Question

If f is one-one onto and differentiable on R. then (f-1)(6)=1/f(6)

Answer 1

False: Because f^-1 here doesn't mean it is the reciprocal of the given function. Rather it is the inverse of the function which means you have to put the certain value in the inverse function you will get pre - image of the function .

Let's see the counter-example.

Suppose f(x)=x², $x\geq 0,$

Then f^-1(x)=√x

Now take the value x=6,we have

f^-1(6)=√6 .....(1)

now f(6)=36

i.e. 1/f(6)=1/36 .....(2)

By equation (1) and (2),

f^-1(6) is NOT equal to 1/f(6).

Answer: The given statement is false.