6well the quesn is like this only!!!:P
prove that the solution(s) of f(x) = inverse of f(x) lie either on the line y = x or on a line whose slope is -1.
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17 Answers
1as the inverse of a curve is it's image about y=x line so if the two curves intersect they will do so on y = x line for sure there would be no case when inverse and original curve intersect and they don't do it on y=x line
The second case is just an additional one ie if the two curves intersect on a line with slope=-1 they will do it in addition to intersecting on y=x line
eg f(x) = x2 , f-1(x) = x1/2
they intersect on (0,0) and (1,1)
case I(y=x)
both points are on y=x
case II(slope = -1)
(0,0) is on y=-x (slope =-1)
(1,1) is on y=2-x (slope =-1)
the image is a bit out of scale
both the cases are simutaneously applicable for the curves which intersect with it's inverse
341The wording of the question is itself so imprecise.
every point lies on a line with slope =-1!
1akhil you didn't read my lines properly...i said that if they do intersect then they will do it on y=x line ...if they don't(as in your figure) then surely they can't intersect on y=x
can you give me any example where a curve and it's inverse are intersecting and they are not doing it on y=x line
39Self-invertible functions like f(x) = 1/x, x ≠0 will not follow this property. It is often wrongly assumed that the number of solutions of the function and its inverse are the number of solutions of the function and line y = x. This is incorrect because self-invertibles possess infinite solutions apart from y = x.
By self-inverse, I mean some functions are their own inverses.
6well it is not so........
in some cases the curve and its image wont intersect at y = x.....
see
1is it so...r u sure about ur second statement...dude thn plz gve me example...
6well what abt the second case then??
its not necessary that all the functions and their inverses intersect at y = x...
1
now carefully observe the equation f(x)=f-1(x) .....u will see tht they will only intersect at y=x..
i hope nw ur doubt is clear
6inverse f(x) means mirror image of graph along y=x line.
slope = -1 means that y=-x
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correct me if am wrong!!!
thinking what else to do...
1the answer is only y=x ...such that the slope is always 1
..
i cannot prove but try examples ...and then check for conditions arising..
341f(x) = 1/x, x≠0 seems to be a counterexample. Any missing conditions?