IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

kislay

If A and B are the points of intersection of y = f(x) and y =f^-1(x) then:
(A)A and B necessarily lie on the line y = x ,
(B)A and B must be coincident
(C)SLope of line AB may be -1
(D)NONE of these
the answer given in book is c and reasoned that if the soln.of f(x)=f^-1(x) , may not lie on y=x.
pls explain

krishna.gopal

Lets take and example
Let f(x)=-sqrt(x) So
finverse(x) = (-x)^2=x^2
A and B are (0,0) and (1,-1) And they have a slope of -1

venkat_tatolu

That is why the answer is MAY BE

kislay

sir if we draw the fraph of the function and its inverse as given by u ,they intersect at only(0,0)

krishna.gopal

You are right Kislay and I am sorry for my answer. Let me try this with a better approach.

We want intersection of y=f(x) and finv(x)

This means

f(x)=finv(x)

or f(f(x))=x

Clearly f(x) = x will be a solution of it so points of intersections can lie on line of slope 1.

It may also lie on a line of slope -1 if we can find a odd function for which we can find a x not equal to zero such that f(x) = -x.

Because then

So f(f(x))=f(-x)

=-f(x) (since function is odd)

=-(-x)=x

And y = -x is one such function.

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