# ​ (a) Let f: R → R be given by ƒ (x) = 2x + 1/3  find ƒoƒ and show that f is invertible.  ​

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(a) Let f: R → R be given by ƒ (x) = $\frac{2x + 1}{3}$ find ƒoƒ and show that f is invertible.

(b) Find the identity element of the binary operation * on N defined by a * b = ab2.

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f of (x) = f(f(x)) = $\frac{2f(x) + 1}{3}$

$\frac{2(\frac{2x+1}{3})+1}{3}$ = $\frac{4x + 5}{9}$

f(x1) = f(x2) ⇒ $\frac{2x_1 + 1}{3 }$ = $\frac{2x_2 + 1}{3 }$

⇒ 3y = 2x +1

⇒ 3y -1 = 2x ⇒ x = $\frac{3y-1}{2}$ ∈ R

Therefore f is onto. Hence f is bijective and invertible.

(b) let ‘e’ be the identity element, then

a*e = a ⇒ ae2 = a ⇒ e2 = 1 ⇒ e = ±1

e*a = a ⇒ ea2 = a ⇒ e = $\frac{1}{a}$

Since e is not unique, this operation has no identity element.