Let f :X $\to$ 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∈ Y, fog 1 (y) = I Y (y) = fog 2 (y). Use one-one ness of f).

Question

Let f :X $\to$  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∈ Y, fog 1 (y) = I Y (y) = fog 2 (y). Use one-one ness of f).

cdquestions Admin · Accepted Answer

Let f : X → Y be an invertible function. Also, suppose f has two inverses (say ).  Then, for all y ∈Y, we have: fog 1 (y)=I y (y)=fog 2 (y) =>f(g 1 (y))=f(g 2 (y)) g 1 (y)=g 2 (y) [f is invertible =>f is one-one =>g 1 = g 2 [g is one-one]. Hence, f has a unique inverse.

	LIST I		LIST II
A.	Range of y=cosec^-1x	I.	R-(-1, 1)
B.	Domain of sec^-1x	II.	(0, π)
C.	Domain of sin^-1x	III.	[-1, 1]
D.	Range of y=cot^-1x	IV.	\([\frac{-π}{2},\frac{π}{2}]\)-{0}

Let f :X\(\to\) Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g₁ and g₂ are two inverses of f. Then for all y∈ Y, fog₁(y) = I_Y(y) = fog₂(y). Use one-one ness of f).

Solution and Explanation

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Questions Asked in CBSE CLASS XII exam

Let f :X\(\to\) Y be an invertible function. Show that f has unique inverse.(Hint: suppose g1 and g2 are two inverses of f. Then for all y∈ Y, fog1(y) = IY (y) = fog2 (y). Use one-one ness of f).

Solution and Explanation

Top Questions on Relations and Functions

Questions Asked in CBSE CLASS XII exam

Let f :X\(\to\) Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g₁ and g₂ are two inverses of f. Then for all y∈ Y, fog₁(y) = I_Y(y) = fog₂(y). Use one-one ness of f).