Page 20 - Mathematics Class - XII

Viva-Voce

1. Is function f defined by f = {(1, 2), (3, 4), (5, 6), (8, 6), (10, –1)} a one to one function?
Ans. Here, two different values in the domain, 5 and 8, have the same output 6, hence function f is not a one to
one function.
2. If X = {1, 2, 3}, Y = {1, 4, 9} and f = {(1, 1), (2, 4), (3, 9)} is a function from X to Y. Is this function
one-one onto?
Ans. Here, every element of X has a distinct image in Y and each element of Y has distinct pre-image in X, so
function f is one-one onto.
3. Find the total number of onto functions (surjective functions) from set X = {1, 2, 3, 4} to set Y = {a, b, c}.
Ans. If X has m elements and Y has n elements, the number of onto functions are,
n – C (n – 1) + C (n – 2) ...(–1) n – 1 n n – 1 (1) ]
[ C
m
n
m
m
m
n
1
2
Here, m = 4 and n = 3,
So, the number of onto functions is: 3 – C (2) + C 1 = 36.
4
3
4
3
4
1 2
4. If X = {a, b, c, d}, Y = {x, y, z} and function f = {(a, z), (b, y), (c, z) and (d, z)}, what type of function is this?
Ans. Here, element x of Y has no pre-image in X. And a, c, d of X has same image z in Y. So, this function is
neither injective nor surjective, i.e., neither one-one nor onto.
5. If : A → B is a bijective function, such that n (A) = 12, then n (B) = ?
Ans. We know that a bijective function is one-one onto function.
So, range of f = Co domain of f
∴ n (A) = n (B) = 12.
MCQs

1. If function f : R → R defined by f (x) = e , then f will be
x
a) one-one but not onto b) one-one and onto
c) Neither one-one nor onto d) None of these

2. Let f : R → R be defined as f (x) = x + 4, choose the correct answer
2
a) one-one but not onto b) one-one onto
c) Neither one-one nor onto d) None of these
3. Function f : N → N defined by f (x) = 7x + 11 is
a) one-one function b) many-one function
c) one-one into function d) many-one onto function
4. Which of the following functions are one-one onto if f : z → z?
a) f (x) = x 3 b) f (x) = x +2 c) f (x) = 2x + 1 d) f (x) = x + 1
2
5. Function f : N → N defined by f (x) = 2x is
a) one-one and onto b) one-one but not onto c) Neither one-one nor onto d) None of these

Answers: 1. a) one-one but not onto 2. b) one-one onto 3. a) one-one function
4. b) f (x) = x +2 5. b) one-one but not onto

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