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A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Let f : X → 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) = 1Y (y) = fog2 (y). Use one-one ness of f)
Solution:
A function is a process or a relation that associates each element 'a' of a non-empty set A, to a single element 'b' of another non-empty set B.
According to the given problem,
Let f : X → Y be an invertible function.
Also, suppose f has two inverses
(g1 and Then, for all y ∈ Y, g2)
fog1 (y) = IY (y) = fog2 (y)
⇒ f (g1 (y)) = f (g2 (y))
⇒ g1 (y) = g2 (y)
[f is invertible ⇒ f is one-one]
⇒ g1 = g2
[g is one-one]
Hence, f has unique inverse
NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.3 Question 10
Let f : X → 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) = 1Y (y) = fog2 (y). Use one-one ness of f)
Summary:
For the function f : X → Y be an invertible function, fog1 (y) = IY (y) = fog2 (y), f is invertible ⇒ f is one-one hence f has unique inverse
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