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Verify Associativity for the Following Three Mappings : F : N → Z0 (The Set of Non-zero Integers), G : Z0 → Q and H : Q → R Given by F(X) = 2x, G(X) = 1/X and H(X) = Ex. - Mathematics

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प्रश्न

Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2xg(x) = 1/x and h(x) = ex.

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उत्तर

Given that f : N → Z0 , g : Z0 → Q and h : Q → R .
gof : N → Q  and hog : Z0 → R

⇒ h o (gof ) : N → R and (hog) o f: N → R
So, both have the same domains.

gof) (x)= g (f (x)) = g (2x) =`1/(2x)`        ...(1)

(hog) (x) = h (g (x)) = h `(1/x) =e^(1/x)`      ...(2)

Now,

( h o (gof))  (x) = h ((gof) (x)) = h `(1/(2x)) = e^(1/(2x))`  [from (1)]

((hog) o f ) (x) = (hog) (f (x)) =  (hog) (2x)  = `e^(1/(2x)`  [from (2)] 

⇒ (h o(gof)) (x) = ((hog) o f) (x), ∀x ∈ N

So, h o (gof)= (hog) o f

Hence, the associative property has been verified

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अध्याय 2: Functions - Exercise 2.2 [पृष्ठ ४६]

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आरडी शर्मा Mathematics [English] Class 12
अध्याय 2 Functions
Exercise 2.2 | Q 9 | पृष्ठ ४६

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