Home
Class 12
MATHS
Let f: X->Ybe an invertible function. Sh...

Let `f: X->Y`be an invertible function. Show that f has unique inverse. (Hint: suppose `g_1( and g)_2`are two inverses of f. Then for all `y in Y ,fog_1(y)=I_Y(y)=fog_2(y)`Use one oneness of f ).

Text Solution

Verified by Experts

Let if possible, g and h be two inverses of `f`.
`therefore `For all `y in Y`
`" " (fog) (y) = 1_Y(y) = (foh) (y)`
`rArr " " f[g(y) ] = f[h(y)]`
`rArr " " g(y) = h(y) " " ( because f` is one-one and onto)
`rArr " " g = h`
Therefore, the inverse of `f` is unique. Hence Proved.
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercise 1.4|13 Videos

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|19 Videos

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercise 1.2|12 Videos

  • PROBABIILITY

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|19 Videos

  • THREE-DIMENSIONAL GEOMETRY

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|23 Videos

Similar Questions

Explore conceptually related problems

Let f: X -> Y be an invertible function. Show that the inverse of f^(-1) is f, i.e., (f^(-1))^(-1)= f .

Let f: X -> Y be an invertible function. Show that the inverse of f^(-1) is f , i.e., (f^(-1))^(-1)= f .

If the function f:A to B is one-one onto and g:B to A , is the inverse of f, then fog =?

Let f:R->R be a function such that f(x+y)=f(x)+f(y),AA x, y in R.

Let Y = {n^2: n in N} in N . Consider f : N ->Y as f(n)=n^2 . Show that f is invertible. Find the inverse of f.

Let Y = {n^2: n in N} in N . Consider f : N ->Y as f(n)=n^2 . Show that f is invertible. Find the inverse of f.

If y=f(x)andx=g(y) are inverse functions of each other, then

If y=f(x)=x^3+x^5 and g is the inverse of f find g^(prime)(2)

Let f(x)=x^(3)+x+1 and let g(x) be its inverse function then equation of the tangent to y=g(x) at x = 3 is

Let f(x) be a differentiable function satisfying f(y)f(x/y)=f(x) AA , x,y in R, y!=0 and f(1)!=0 , f'(1)=3 then