Calculus CG term 1 Revision

1. 1Let f be a rule assigning elements of the set Y to elements of the set X. We say that f is well-defined or unambiguous if and only if

   1. Af(x) = f(x′) whenever x = x′
   2. Bf(x) = f(x′) whenever x = x
   3. Cf(x) = f(x) whenever x = x′
   4. Dg(x) = f(x′) whenever x = x′
   5. Ef(y) = f(x′) whenever y = x′
2. 2Definition 1.2. A function f from X to Y consist of three things:
   (1) a set X = dom(f) called the [blank_start]domain[blank_end] of f;
   (2) a set Y = codom(f) called the [blank_start]co-domain[blank_end] of f;
   (3) a well-defined rule that assigns a unique element [blank_start]f(x)[blank_end] ∈ Y to each x ∈ X.

   1. Adomain
   2. Bcodomain
   3. Cfunction
   4. Dexpression
   5. Eco-domain
   6. Fdomain
   7. Gfunction
   8. Hequation
   9. Iset
   10. Jf(x)
   11. Kx
   12. Ln
   13. Min the domain
3. 3If X ⊆ R, we say that f is a function of

   1. AA Real variable
   2. BA Series
   3. CA Natural number
   4. DA Number
   5. EX
4. 4 If Y ⊆ R, we say that f is

   1. AA real value function
   2. BA natural value function
   3. CA member of Y
   4. DA set
   5. EA subset
5. 5If X ⊆ R and Y ⊆ R, then we say that f is

   1. Aa real-valued function of a real variable.
   2. Ba real-valued function of a variable.
   3. Ca valued function of a real variable.
   4. Da real-valued equation of a real variable.
   5. Ea real-valued function of a real variale.
   6. Fa function of a real variable.
6. 6Let f : X → Y be a function. The range of f is the set

   1. Aran ( f ) = { y ∈ Y : y = f ( x ) for some x ∈ X }.
   2. Bran ( f ) = { y ∈ Y : y = f ( x ) for all x ∈ X }.
   3. Cran ( f ) = { y ∈ Y : for some x ∈ X }.
   4. Dran ( f ) = { y ∈ Y : y = x for some x ∈ X }.
   5. Eran ( y ) = { y ∈ Y : y = f ( x ) for some x ∈ X }.
   6. Fran ( f ) = { x ∈ Y : y = f ( x ) for some x ∈ X }.
   7. Gran ( f ) = { y ∈ X : y = f ( x ) for some x ∈ X }.
   8. Hran ( f ) = { y ∈ Y : y = f ( x ) }.
7. 7Two functions f and g are said to be equal if and only if:
   
   (1) dom ( f ) = dom ( g ),
   (2) codom ( f ) = codom ( g ), and
   (3) for all x ∈ X, f ( x ) = g ( x ).

   1. AVerdadero
   2. BFalso
8. 8Two functions f and g are said to be equal if and only if:
   
   (1) dom ( f ) = dom ( g ),
   (2) codom ( f ) = codom ( g ), and
   (3) for some x ∈ X, f ( x ) = g ( x ).

   1. AVerdadero
   2. BFalso
9. 9Two functions f and g are said to be equal if and only if:
   
   (1) dom ( f ) = codom ( g ),
   (2) codom ( f ) = dom ( g ), and
   (3) for all x ∈ X, f ( x ) = g ( x ).

   1. AVerdadero
   2. BFalso
10. 10Two functions f and g are said to be equal if and only if:
    
    (1) dom ( f ) = dom ( f ),
    (2) codom ( f ) = codom ( g ), and
    (3) for all x ∈ X, f ( x ) = g ( x ).

    1. AVerdadero
    2. BFalso

11. 11Two functions f and g are said to be equal if and only if:
    
    (1) dom ( f ) = dom ( g ),
    (2) codom ( f ) = codom ( g ), and
    (3) for all x ∈ Y, f ( x ) = g ( x ).

    1. AVerdadero
    2. BFalso
12. 12Two functions f and g are said to be equal if and only if:
    
    (1) dom ( f ) = dom ( g ),
    (2) codom ( f ) = codom ( g ), and
    (3) for all y ∈ X, f ( x ) = g ( x ).

    1. AVerdadero
    2. BFalso
13. 13The composition of the functions f and g is the function g ◦ f : X → Z defined by the rule g ◦ f ( x ) = g

    1. AVerdadero
    2. BFalso
14. 14The composition of the functions f and g is the function g ◦ f : X → Z defined by the rule g ◦ f ( x ) = f ( x )

    1. AVerdadero
    2. BFalso
15. 15The composition of the functions f and g is the function g ◦ f : X → Z defined by the rule f ◦ f ( x ) = g

    1. AVerdadero
    2. BFalso
16. 16The composition of the functions f and g is the function f ◦ f : X → Z defined by the rule g ◦ f ( x ) = g

    1. AVerdadero
    2. BFalso
17. 17The addition of the functions f and g is the function g ◦ f : X → Z defined by the rule g ◦ f ( x ) = g

    1. AVerdadero
    2. BFalso
18. 18The multiplication of the functions f and g is the function g ◦ f : X → Z defined by the rule g ◦ f ( x ) = g

    1. AVerdadero
    2. BFalso
19. 19If f : X → X and f(x) = x for all x, then f is called

    1. AIdentity function
    2. BConstant function
20. 20If f : X → Y and f(x) = y for all x ∈ X, then f is often called

    1. Aconstant function y.
    2. BIdentity function y.
21. 21Two functions f and g are said to be equal if and only if:

    1. A(1) dom( f ) = dom( g ),
       (2) codom ( f ) = codom( g ), and
       (3) for all x ∈ X , f ( x ) = g ( x ).
    2. B(1) dom( f ) = dom( g ),
       (2) codom ( f ) = codom( g ), and
       (3) for some x ∈ X , f ( x ) = g ( x ).
    3. C(1) dom( f ) = dom( g ),
       (2) codom ( f ) = codom( g ), and
       (3) for all x ∈ X , f ( x ) = f ( x ).
    4. D(1) dom( f ) = dom( g ),
       (2) codom ( f ) = codom( g ), and
       (3) for all x ∈ X , f ( x ) = fg ( x ).
    5. E(1) dom( f ) = dom( g ),
       (2) codom ( f ) = codom( g ), and
       (3) for all y ∈ X , f ( x ) = g ( x ).
    6. F(1) dom( f ) = dom( g ),
       (2) codom ( f ) = codom( g ), and
       (3) for all x ∈ X , g ( x ) = g ( x ).
    7. G(1) codom( f ) = dom( g ),
       (2) codom ( f ) = codom( g ), and
       (3) for all x ∈ X , f ( x ) = g ( x ).
    8. H(1) dom( f ) = codom( g ),
       (2) codom ( f ) = codom( g ), and
       (3) for all x ∈ X , f ( x ) = g ( x ).
    9. I(1) dom( f ) = dom( g ),
       (2) codom ( f ) = dom( g ), and
       (3) for all x ∈ X , f ( x ) = g ( x ).
    10. J(1) dom( f ) = dom( g ),
        (2) codom ( f ) = codom( g ), and
        (3) for all x ∈ Y , f ( x ) = g ( x ).
22. 22The range of f is the set

    1. Aran ( f ) = { y ∈ Y :y = f ( x ) for some x ∈ X }.
    2. Bran ( f ) = { y ∈ Y :y = f ( x ) for all x ∈ X }.
    3. Cran ( f ) = { x ∈ X :y = f ( x ) for some x ∈ X }.
    4. Dran ( f ) = { y ∈ X :y = f ( x ) for some x ∈ X }.
    5. Eran ( f ) = { y ∈ Y :y = f ( x ) for some x ∈ Y }.
    6. Fran ( f ) = { y ∈ Y :y = f ( x ) for some x ∈ R }.
    7. Gran ( f ) = { y ∈ Y :y = f ( x ) for some y}.
23. 23The inverse of f, denoted f−1, if it
    exists, is the function f−1 : Y → X with the property that

    1. Af(x) = y if and only if f−1(y) = x.
    2. Bf(x) = y and only if f−1(y) = x.
    3. Cf(x) = y if and only if f−1(x') = x.
24. 24surjective

    1. Aif and only if for every y ∈ Y , there is at least one x ∈ X such that f(x) = y.
    2. Bif and only if for every y ∈ Y , there is x ∈ X such that f(x) = y.
    3. Cif and only if for every y ∈ Y ,there is x ∈ X such that f(x) = y.
    4. Dif and only if for some y ∈ Y , there is at least one x ∈ X such that f(x) = y.
    5. Eif and only if for every x ∈ X , there is at least one y ∈ Y such that f(x) = y.
    6. Fif and only if for every x ∈ X , there is y ∈ Y such that f(x) = y.
    7. Gif and only if for some x ∈ X , there is at least one y ∈ Y such that f(x) = y.
25. 25Which of the following mean the same thing

    1. ASurjective
    2. BInjective
    3. CBijective
    4. DSuggestive
    5. EInjestive
    6. FOnto
26. 26Another word for onto

    1. ASurjective
    2. BInjective
    3. CBijective
27. 27Another word for surjective

    1. AOnto
    2. BOne-to-one
28. 28Which of the following mean the same

    1. Ainjective
    2. Bone to one
    3. Cbijection
    4. Donto
    5. Einjection
29. 29 injective

    1. Aif and only if for every y ∈ Y , there is at most one x ∈ X such that f(x) = y.
    2. Bif and only if for every y ∈ Y , there is more than one x ∈ X such that f(x) = y.
    3. Cif and only if for some y ∈ Y , there is at most one x ∈ X such that f(x) = y.
    4. Devery y ∈ Y , there is at most one x ∈ X such that f(x) = y.
    5. Eif and only if for every y ∈ Y , there is at most none x ∈ X such that f(x) = y.
    6. Fif and only if for every y ∈ Y , there is at most one x ∈ X such that f(x) = o.
    7. Gif and only if for every x ∈ X , there is at most one x ∈ X such that f(x) = y.
    8. Hif and only if for some x ∈ X , there is at most one x ∈ X such that f(x) = y.
    9. Iif and only if for every x ∈ X , there is at most one y ∈ X such that f(x) = y.
30. 30bijective

    1. Aif and only if f is both one-to-one and onto
    2. Bif and only if f is one-to-one
    3. Cif and only if f is onto
    4. Done-to-one and onto
    5. Eone-too-one and onto
31. 31Let f : X → Y be a function. The following are equivalent:

    1. Af is one-to-one;
    2. Bif x ̸= x′, then f(x) ̸= f(x′);
    3. Cif f(x) = f(x′), then x = x′.
    4. Dif f(x) = f(x′), then x /= x′.
    5. Eif x ̸= x′, then f(x)= f(x′);
    6. Fif x= x′, then f(x) ̸= f(x′);
    7. Gif x ̸= x′, then f(x) ̸= f(x);
32. 32Let f : X → Y be a function.
    
    f is one to one

    1. AVerdadero
    2. BFalso
33. 33Let f : X → Y be a function.
    f is onto

    1. AVerdadero
    2. BFalso
34. 34Let f : X → Y be a function.
    
    f is not injective

    1. AVerdadero
    2. BFalso
35. 35Let f : X → Y be a function.
    
    f is onto if and only if ran(f) = codom(f).

    1. AVerdadero
    2. BFalso
36. 36Let f : X → Y be a function.
    
    f is onto if and only if dom(f) = codom(f).

    1. AVerdadero
    2. BFalso
37. 37Let f : X → Y be a function.
    
    f is one-to-one if and only if ran(f) = codom(f).

    1. AVerdadero
    2. BFalso
38. 38f is a bijection if and only if for every y ∈ Y there is exactly one x ∈ X such
    that f(x) = y.

    1. AVerdadero
    2. BFalso
39. 39f is a bijection if and only if for some y ∈ Y there is exactly one x ∈ X such
    that f(x) = y.

    1. AVerdadero
    2. BFalso
40. 40f is a bijection if and only if for every y ∈ Y there is more than one x ∈ X such
    that f(x) = y.

    1. AVerdadero
    2. BFalso
41. 41f is a surjection if and only if for every y ∈ Y there is exactly one x ∈ X such
    that f(x) = y.

    1. AVerdadero
    2. BFalso
42. 42Let f:X→Y be a function.
    (1) f has an inverse if and only if it is a bijection.
    (2) If f is a bijection, then f ◦ f−1 is the identity function on Y and f−1 ◦ f is the
    identity function on X.

    1. AVerdadero
    2. BFalso
43. 43Let f : X→Y be a function.

    1. Af has an inverse if and only if it is a bijection
    2. Bf has an inverse if and only if it is a surjection
    3. Cf has an inverse if and only if it is an injection
    4. Df has an inverse if and only if it is a real function
44. 44 f is one-to-one if and only if every horizontal line crosses the graph of f at most once.

    1. AVerdadero
    2. BFalso
45. 45 f is one-to-one if and only if every horizontal line crosses the graph of f.

    1. AVerdadero
    2. BFalso
46. 46 f is one-to-one if and only if every horizontal line crosses the graph of f more than once.

    1. AVerdadero
    2. BFalso
47. 47 f is real if and only if every horizontal line crosses the graph of f at most once.

    1. AVerdadero
    2. BFalso
48. 48 f is one-to-one if and only if every vertical line crosses the graph of f at most once.

    1. AVerdadero
    2. BFalso
49. 49f is onto if and only if every horizontal line crosses the graph of f at least once.

    1. AVerdadero
    2. BFalso
50. 50f is onto if and only if every horizontal line crosses the graph of f at most once.

    1. AVerdadero
    2. BFalso
51. 51f is onto if and only if every vertical line crosses the graph of f at least once.

    1. AVerdadero
    2. BFalso
52. 52f is onto if and only if every horizontal line crosses the graph of f at least once.

    1. AVerdadero
    2. BFalso
53. 53f is a bijection if and only if every horizontal line crosses the graph of f exactly once.

    1. AVerdadero
    2. BFalso
54. 54f is a bijection if and only if some horizontal line crosses the graph of f exactly once.

    1. AVerdadero
    2. BFalso
55. 55f is a bijection if and only if every horizontal line crosses the graph of f at least once.

    1. AVerdadero
    2. BFalso
56. 56f is a bijection if and only if every horizontal line crosses the graph of f at most once.

    1. AVerdadero
    2. BFalso
57. 57Strictly speaking a function is not properly defined unless the domain, co-domain and the rule defining it have been specified.

    1. AVerdadero
    2. BFalso
58. 58Strictly speaking a function is properly defined unless the domain, co-domain and the rule defining it have been specified.

    1. AVerdadero
    2. BFalso
59. 59Strictly speaking a function is not properly defined unless the domain and the co-domain have been specified.

    1. AVerdadero
    2. BFalso
60. 60Strictly speaking a function is not properly defined unless the co-domain and the rule defining it have been specified.

    1. AVerdadero
    2. BFalso
61. 61Strictly speaking a function is not properly defined unless the .........., ............ and the .............. defining it have been specified.

    1. ACodomain
    2. BDomain
    3. CRule
    4. DBijection
    5. ESummation
    6. FExpression
    7. GValue
62. 62domain convention.
    

    1. Aassume that the co-domain of f is R and that the domain of f is the largest set of real numbers for which the rule is valid.
    2. Bassume that the domain of f is R and that the domain of f is the largest set of real numbers for which the rule is valid.
    3. Cassume that the co-domain of f is R and that the domain of f is the largest set of real numbers.
    4. Dassume that the co-domain of f is R and that the domain of f is R.
    5. Eassume that the co-domain of f is R and that the domain of f is Z.
63. 63periodic with period p if, for each x ∈ R,

    1. Af(x + p) = f(x)
    2. Bf(x + p) = f(x) + p
    3. Cf(x + p) = x + p
    4. Df(x + p) = y + p
    5. Ef(x + p) = f(x) + y
    6. Ff(x + p) = f(y) + p
    7. Gf(p) = f(x)
    8. Hf(p) = f(x) + p
    9. If(p) = f(x + p)
    10. Jx = p
64. 64If a function f is periodic, the smallest positive number p s.t. f(x + p) = f(x) for all x ∈ X is called the fundamental period of f.
    

    1. AVerdadero
    2. BFalso
65. 65If a function f is periodic, the smallest positive number p s.t. f(x + p) = f(x) for all x ∈ X is called the
    

    1. Afundamental period of f.
    2. Bperiod of f.
    3. Cfundamental period of x.
    4. Dperiod of x.
66. 66If a function f is periodic, the smallest positive number p s.t. f(x + p) = f(x)
    for all x ∈ X is called the [blank_start]fundamental period[blank_end] of [blank_start]f[blank_end].

    1. Afundamental period
    2. Bf
67. 67f is said to be even if and only if

    1. Af(−x) = f(x) for all x ∈ R.
    2. Bf(x) = f(x) for all x ∈ R.
    3. Cf(x) = -f(x) for all x ∈ R.
    4. Df(-x) = f(-x) for all x ∈ R.
    5. Ef(x) = f(x) for all x ∈ N.
    6. Ff(−x) = f(x) for all x ∈ N.
    7. Gf(−x) = f(x) for some x ∈ R.
68. 68f is said to be odd if and only if

    1. Af(−x) = −f(x) for all x ∈ R.
    2. Bf(x) = −f(x) for all x ∈ R.
    3. Cf(−x) = f(x) for all x ∈ R.
    4. Df(−x) = −f(x) for some x ∈ R.
    5. Ef(−x) = f(x) for some x ∈ R.
69. 69f is said to be odd if and only if f(−x) = −f(x) for all x ∈ R.

    1. AVerdadero
    2. BFalso
70. 70f is said to be odd if and only if f(x) = −f(x) for all x ∈ R.

    1. AVerdadero
    2. BFalso
71. 71f is said to be even if and only if f(-x) = f(x) for all x ∈ R.

    1. AVerdadero
    2. BFalso
72. 72f is said to be even if and only if f(x) = -f(x) for all x ∈ R.

    1. AVerdadero
    2. BFalso
73. 73f is said to be even if and only if f(-x) = f(x) for x ∈ R.

    1. AVerdadero
    2. BFalso

Descripción

Bachelors Degree Mathematics Test sobre Calculus CG term 1 Revision, creado por Will Rickard el 28/10/2015.

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Will Rickard

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