Functions one-one into and onto - Question Bank

1. Which condition must be met for a function f: A → B to be classified as 'into'?
A) Range(f) = B
B) Range(f) ⊂ B and Range(f) ≠ B
C) Range(f) = A
D) Range(f) ⊃ B
2. A function f: A → B is called a bijection if it is:
A) Either one-one or onto
B) Both one-one and into
C) Both into and onto
D) Both one-one and onto
3. Let f(x) = x² for f: {-1, 1} → {1}. Is this function onto?
A) Yes, because the range {1} is equal to the codomain {1}.
B) No, because f is not one-one.
C) Yes, because the domain is {-1, 1}.
D) No, because the domain has more elements than the codomain.
4. Let f(x) = x² for f: {-1, 1} → {1}. Is this function one-one?
A) Yes, because the range has only one element.
B) No, because f(-1) = f(1) = 1.
C) Yes, because -1 ≠ 1.
D) No, because the domain has more than one element.
5. If f: A → B is a function, and A = {1, 2}, B = {3}. Is this function onto?
A) Yes, because the only element in B is mapped to.
B) No, because |A| > |B|.
C) Yes, because it is one-one.
D) No, because it is not one-one.
6. If f: A → B is a function, and A = {1, 2}, B = {3}. Is this function one-one?
A) Yes, because |A| > |B| is not true.
B) No, because |A| > |B|.
C) Yes, because it's the only possible function.
D) No, because two elements must map to the same element.
7. If f: A → B is a function, and A = {1, 2}, B = {3}. How many functions are possible?
A) 1
B) 2
C) 3
D) 4
8. What is the total number of onto functions from a set A with m elements to a set B with n elements, where m ≥ n?
A) nᵐ
B) P(n, m)
C) n! S(m, n)
D) m! S(n, m)
9. What is the total number of one-one functions from a set A with m elements to a set B with n elements, where m ≤ n?
A) n!
B) P(n, m)
C) C(n, m)
D) nᵐ
10. What is the total number of functions from a set A with m elements to a set B with n elements?
A) mⁿ
B) nᵐ
C) m + n
D) m * n
11. If f: R → R is defined by f(x) = 1/x, is this function onto?
A) Yes, because for every y in R, x = 1/y.
B) No, because 0 is in the codomain but not in the range.
C) Yes, because it is one-one.
D) No, because x cannot be 0.
12. If f: R → R is defined by f(x) = 1/x, is this function one-one?
A) Yes, because 1/x is unique for each x.
B) No, because 1/0 is undefined.
C) Yes, because it is defined for all non-zero reals.
D) No, because f(x) = f(-x) is not true.
13. For a function f: A → B to be one-one, the inverse image of each element in the codomain must be:
A) Either empty or a single element
B) A single element
C) Empty
D) Multiple elements
14. If f: A → B is a function and there exists an element b ∈ B such that f(x) ≠ b for all x ∈ A, then f is:
A) One-one
B) Onto
C) Into
D) Bijective
15. Which of the following describes a function that is neither one-one nor onto?
A) f(x) = x, f: R → R
B) f(x) = 2x, f: R → R
C) f(x) = x², f: R → R
D) f(x) = 5, f: R → R
16. Let f(x) = eˣ, where f: R → R. Is this function onto?
A) Yes, because eˣ can take any real value.
B) No, because the range is (0, ∞), not R.
C) Yes, because it is one-one.
D) No, because the domain is R.
17. Let f(x) = eˣ, where f: R → R. Is this function one-one?
A) Yes, because eˣ is always positive.
B) No, because eˣ can take the same value for different x.
C) Yes, because it is strictly increasing.
D) No, because the range is not R.
18. If f: A → B is a function such that |A| = n and |B| = m. If f is onto, then:
A) n > m
B) n < m
C) n = m
D) n ≥ m
19. If f: A → B is a function such that |A| = n and |B| = m. If f is one-one, then:
A) n > m
B) n < m
C) n = m
D) n ≤ m
20. Consider the function f(x) = x mod 3, where f: Z → {0, 1, 2}. Is this function one-one?
A) Yes, because x mod 3 is unique for each x.
B) No, because f(0) = f(3) = 0.
C) Yes, because the codomain is finite.
D) No, because the domain is infinite.
21. Consider the function f(x) = x mod 3, where f: Z → {0, 1, 2}. Is this function onto?
A) Yes, because every element in the codomain is obtained.
B) No, because multiple elements in Z map to the same value.
C) Yes, because it is a well-defined function.
D) No, because the domain is infinite.
22. If f: A → B is an onto function, then which of the following is true?
A) f⁻¹(y) is defined for all y ∈ B
B) f⁻¹(y) is defined for some y ∈ B
C) f⁻¹(y) is never defined
D) f⁻¹(y) is defined only if f is one-one
23. Let A = {1, 2} and B = {3, 4}. How many onto functions can be defined from A to B?
A) 0
B) 1
C) 2
D) 4
24. Let A = {1, 2} and B = {3, 4}. How many one-one functions can be defined from A to B?
A) 1
B) 2
C) 3
D) 4
25. If f: A → B is one-one, then the number of elements in A must be:
A) Greater than the number of elements in B
B) Less than the number of elements in B
C) Equal to the number of elements in B
D) Less than or equal to the number of elements in B
26. A function f: A → B is called a(n) ______ function if every element of B has exactly one pre-image in A.
A) Into
B) One-one
C) Onto
D) Bijective
27. Let f(x) = 2ˣ, where f: R → R. Is this function onto?
A) Yes, because 2ˣ can be any real number.
B) No, because the range is (0, ∞), not R.
C) Yes, because it is strictly increasing.
D) No, because it is not one-one.
28. If f: A → B is one-one and |A| = |B|, then f must be:
A) Into
B) Onto
C) Bijective
D) Neither onto nor into
29. Consider the function f(x) = x+1 for f: Z → Z. Is this function onto?
A) Yes, because for every y in Z, we can find x = y-1 in Z.
B) No, because x must be positive.
C) Yes, because it is one-one.
D) No, because the range is Z+.
30. Consider the function f(x) = x+1 for f: Z → Z. Is this function one-one?
A) Yes, because for every y in Z, there is a unique x in Z such that f(x) = y.
B) No, because f(x) = f(x+1).
C) Yes, because it is a linear function.
D) No, because the domain and codomain are integers.
31. If a function f: A → B is not onto, it means:
A) There exists at least one element in B that is not the image of any element in A.
B) There exists at least one element in A that is not mapped to any element in B.
C) Every element in B is mapped by multiple elements in A.
D) No element in A maps to the same element in B.
32. Let f(x) = sin(x), where f: R → R. Is this function onto?
A) Yes, because sin(x) covers all values between -1 and 1.
B) No, because the range is [-1, 1], not R.
C) Yes, because it is periodic.
D) No, because it is not one-one.
33. What is the minimum number of elements required in the codomain B for a function f: A → B to be onto, if |A| = 5?
A) 1
B) 3
C) 5
D) Cannot be determined
34. If f: A → B is one-one, then for any two distinct elements x₁, x₂ ∈ A, we have:
A) f(x₁) = f(x₂)
B) f(x₁) ≠ f(x₂)
C) f(x₁) < f(x₂)
D) f(x₁) > f(x₂)
35. Consider f(x) = x³ where f: R → R. Is this function onto?
A) Yes, because x³ can take any real value.
B) No, because x³ is always positive.
C) Yes, because it is one-one.
D) No, because its range is restricted.
36. If f(x) = x for all x in A, where f: A → A, this is called:
A) Zero function
B) Identity function
C) Constant function
D) Inclusion function
37. If f(x) = c (a constant), where f: R → R, is this function one-one?
A) Yes, if the domain has only one element.
B) No, unless the domain has only one element.
C) Yes, always.
D) No, never.
38. Which type of function guarantees that every element in the codomain has at least one pre-image in the domain?
A) Into function
B) One-one function
C) Onto function
D) Constant function
39. For a function f: A → B, if f is onto, what can be said about the number of elements in the range and codomain?
A) Number of elements in Range(f) < Number of elements in B
B) Number of elements in Range(f) > Number of elements in B
C) Number of elements in Range(f) = Number of elements in B
D) Number of elements in Range(f) ≤ Number of elements in B
40. For a function f: A → B, if |A| = m and |B| = n, and f is one-one, what can be said about m and n?
A) m < n
B) m > n
C) m = n
D) m ≤ n
41. Let f(x) = |x|, where f: R → R. Is this function onto?
A) Yes, because the range is all real numbers.
B) No, because negative numbers are not in the range.
C) Yes, because it is a continuous function.
D) No, because it is not one-one.
42. If A = {1, 2, 3} and B = {a, b, c, d}, and f: A → B is defined as f(1)=a, f(2)=b, f(3)=c. Is f an onto function?
A) Yes, because all elements of A are mapped.
B) No, because 'd' in B is not mapped.
C) Yes, because f is one-one.
D) No, because the number of elements in A is less than in B.
43. Consider the function f(x) = x², where f: R → R. Is this function onto?
A) Yes, because the range includes all positive numbers.
B) No, because negative numbers are not in the range.
C) Yes, because the domain is R.
D) No, because it is not one-one.
44. Consider the function f(x) = x², where f: R → R. Is this function one-one?
A) Yes, because x² is always positive.
B) No, because f(x) = f(-x) for non-zero x.
C) Yes, because it is a quadratic function.
D) No, because the range is not R.
45. Consider the function f(x) = 2x + 3, where f: R → R. Is this function one-one?
A) Yes, because for every y in R, there is a unique x in R such that f(x) = y.
B) No, because different x values can map to the same y value.
C) Yes, because it is a linear function with a non-zero slope.
D) No, because the domain and codomain are the same.
46. Which of the following statements is true for an 'into' function f: A → B?
A) Range(f) = B
B) Range(f) ⊂ B
C) Range(f) = A
D) Range(f) ⊃ B
47. If f: A → B is both one-one and onto, then it is called a(n):
A) Into function
B) Bijective function
C) Constant function
D) Identity function
48. A function f: A → B is called onto (surjective) if:
A) Each element in A maps to a unique element in B.
B) The image of every element in A is distinct.
C) The range of f is equal to the codomain B.
D) The function is not one-one.
49. What is the condition for a function f: A → B to be one-one (injective)?
A) Every element of B is mapped by at least one element of A.
B) Distinct elements of A are mapped to distinct elements of B.
C) Every element of A is mapped to exactly one element of B.
D) The range of f is a proper subset of the codomain B.
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