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. Let f : R → R be defined as f(x) = 3x. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.

Class 12, NCERT Chapter 1,  Exercise 1.2, Q12 f : R → R defined as  f ( x ) = 3 x . Let  x ,  y  ∈  R  such that  f ( x ) =  f ( y ). ⇒ 3 x  = 3 y ⇒  x  =  y ∴ f  is one-one. Also, for any real number ( y)  in co-domain  R , there exists  in  R  such that . ∴ f  is onto. Hence, function  f  is one-one and onto. The correct answer is A.
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Let f : R → R be defined as f(x) = x 4 . Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.

Class 12, NCERT Chapter 1,  Exercise 1.2, Q11 f : R → R is defined as  f (x) = x 4 Let  x ,  y  ∈  R  such that  f ( x ) =  f ( y ). ∴ does not imply that x 1 =x 2 . For instance, f (1) =  f (-1) = 1 ∴  f  is not one-one. Consider an element 2 in co-domain  R . It is clear that there does not exist any  x  in domain  R  such that  f ( x ) = 2. ∴  f  is not onto. Hence, function  f  is neither one-one nor onto. The correct answer is D
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Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by f(x) = (x-2)/(x-3) Is f one-one and onto? Justify your answer.

Class 12, NCERT Chapter 1,  Exercise 1.2, Q10 A =  R  - {3}, B =  R  - {1} f : A → B is defined as Let x,y∈A such that f(x)=f(y). ⇒ (x-2)/(x-3)=(y-2)/(y-3) ⇒ (x-2)(y-3)=(y-2)(x-3) ⇒ xy-3x-2y+6=xy-3y-2x+6 ⇒ -3x-2y=-3y-2x ⇒ 3x-2x=3y-2y ⇒ x=y ∴  f  is one-one. Let  y  ∈B =  R  - {1}. Then,  y  ≠ 1. The function  f  is onto if there exists  x  ∈A such that  f ( x ) =  y. Now, f(x)=y ⇒(x-2)(x-3)=y ⇒x-2=xy-3y ⇒x(1-y)=-3y+2 ⇒ x=2-3y/1-y  ∈A   Thus, for any  y  ∈ B, there exists  x=2-3y/1-y  ∈A     such that Hence, function  f  is one-one and onto.
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Let f : N → N be defined by f (n) ={ (n+1)/2, if n is odd and (n-1)/2, if n is even, for all n ∈ N.State whether the function f is bijective. Justify your answer

Class 12, NCERT Chapter 1,  Exercise 1.2, Q9 Thus it is bijective.
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In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f : R → R defined by f(x) = 3 – 4x (ii) f : R → R defined by f(x) = 1 + x^2

Class 12, NCERT Chapter 1,  Exercise 1.2, Q7 (i)  Here f(x) = 3 – 4x Then f(x 1 ) = 3 – 4x 1 and f (x 2 ) = 3 – 4x 2 Now f(x 1 ) = f (x 2 ) ⇒ 3 – 4x 1  = 3 – 4x 2 ⇒ – 4x 1  = –4x 2 ⇒ x 1  = x 2 ∴ f(x 1 ) = f(x 2 ) & x 1  = x 2 So, f is one-one function. Let f(x) = y ∈ R Then y=3 - 4x ⇒ x= 3-y/4 (ii) Let x 1  = 2 and x 2  = –2 ∈ R Here f(x) = 1 + x 2 Then f(x 1 ) = f (2) = 1 + (2)  2  = 5 and f (x 2 ) = f(–2) = 1 + (–2)  2  = 5 ∴ f (x 1 ) = f(x 2 ) but x 1  ≠ x 2 So, f is not one-one function. Let f(x) = – 2 ⇒R Then 1 + x 2  = –2 ⇒ x 2  = –3 = ± √–3 ∈R So, f is not onto function.
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Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.

Class 12, NCERT Chapter 1,  Exercise 1.2, Q6 It is given that A={1,2,3} B={4,5,6,7} f : A →B defined as f (x)=  {(1, 4), (2, 5), (3, 6)}  Therefore  f(1)=4,   f(2)=5,   f(3)=6 It is seen that the images of distinct elements of A under  f  are distinct. Hence, function  f  is one-one
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