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|Exercise Exercise 1.4|13 Videos

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN|Exercise Miscellaneous Exercise|19 Videos

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN|Exercise Exercise 1.2|12 Videos

  • PROBABIILITY

    NAGEEN PRAKASHAN|Exercise Miscellaneous Exercise|19 Videos

  • THREE-DIMENSIONAL GEOMETRY

    NAGEEN PRAKASHAN|Exercise Miscellaneous Exercise|23 Videos

Similar Questions

Explore conceptually related problems

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

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

If g (y) is inverse of function f :R to R given by f (x)=x+3, then g (y)=

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

Let y=f(x) and y=g(x) be two monotonic, bijective functions such that f(g(x) is defined than y=f(g(x)) is

Let function f:X rarr Y, defined as f(x)=x^(2)-4x+5 is invertible and its inverse is f^(-1)(x), then