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(i) Consider ƒ : R → R given by ƒ(x) = 5x + 2 

(a) Show that f is one-one. 

(b) Is f invertible? Justify your answer. 

(ii) Let * be a binary operation on N defined by a * b = HCF of a and b 

(a) Is * commutative? 

(b) Is * associative?

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(i) (a) Let x , x , ∈ R 

ƒ(x1) = ƒ(x2) ⇒ 5x1 + 2 

= 5x2 + 2 

⇒ 5x2 = 5x2 ⇒ x1 = x2 

Therefore s one-one.

(b) Yes. 

Let y e range of ƒ 

⇒ ƒ(x) = y ⇒ 5x + 2 = y

⇒ x = \(\frac{y-2}{5}\) ∈ R

Therefore corresponding to every y ∈ R there existsa real number \(\frac{y-2}{5}\) Therefore f is onto. 

Hence bijective, so invertible.

(ii) (a) Yes. 

a * b = HCF (a,b) = HCF (b,a) = b * a 

Hence commutative.

(b) Yes. 

a * (b * c) = a* HF(b,c) = HCF(a,b,c) 

(a*b) * c =HCF(a,b) * c HCF(a,b,c) 

a * (b * c) = (a * b) * c

 Hence associative.

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