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1st year Maths > RAC functions > Flashcards

RAC functions Flashcards

1
Q

function

A

let A and B be sets. A function from A to B is a rule that associates every element of A to exactly one element of B.

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2
Q

set of inputs

A

domain

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3
Q

set of outputs

A

codomain

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4
Q

image/range

A

for every element from A, xЄA then the element of B associated to x by f is called the image

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5
Q

difference between codomain and image/range

A

codomain are the possible outputs

range/image are the actual outputs

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6
Q

image definition (function f: A->B)

A

the subset of B whose elements are the images of elements A. denoted by F(A)

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7
Q

f: A->B g: C->D f=g if…

A
  1. A=C
  2. B=D
  3. f(x) = g(x) for all xЄA
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8
Q

graph of symbol

A

Γ

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9
Q

image letters

A

f(A) = {f(x): xЄA}

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10
Q

graph letters

A

Γf = {(x,y)ЄAxB: y=f(x)}

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11
Q

how to tell if a graph is a graph of a function

A

if one input has multiple outputs it is not a function (draw a vertical line, if it crosses two points its not the graph of a fucntion)

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12
Q

constructing function via restriction definintion

A

Let f:A->B. Let S⊆A. The restriction of f to S is the function f|s: S->B defined by f|s(x) = f(x) for all values of xЄS

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13
Q

what is restriction

A

taking a smaller domain to only show part of the function

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14
Q

constructing a function via composition definition

A

Let f:A->B and g:C->. Assume that f(A)⊆C (if B=C then definitely true). Then the composition of g and f is the function g∘f: A-> D defined by g∘f(x) = g(f(x)) for all values of xЄA

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15
Q

indentity function definition

A

Let A be a set. The identity function of A is the function idₐ: A->A defined by idₐ(x) = x for all values of xЄA

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16
Q

what happens if you do composition with an identity function

A

nothing

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17
Q

Constant Function definition

A

A function f:A->B is said to be constant if there exists bЄB such that f(x)=b for all xЄA

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18
Q

what do functions have to be to be invertible

A

one-to-one

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19
Q

invertibility definition

A

Let f: A-> B. We say that f is invertible if there exists a function g:B->A such that for all xЄA and yЄB f(x) = y <=> g(y) =x

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20
Q

Injective (one-to-one)

A

if for all x,x’ЄA: x=x’ => f(x)!=f(x’) (all points have distinct images)
if every image point has at most one preimage point in A.

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21
Q

surjective

A

f(A) = B

if every image point has at least one preimage in A

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22
Q

bijective

A

both surjective and injective.

if every image point has exactly one preimage in A

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23
Q

g: B->A is the inverse of f:A->B iff

A

g∘f = idₐ and f∘g = idb

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24
Q

real valued function of a real variable

A

f: A->ℝ , where A⊆ℝ

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25
Q

domain convention

A
  • assume codomain is ℝ

- the domain is the largest subset of ℝ where the given subset makes sense

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26
Q

finding the real valued inverse

A

(the function f: A-> B is injective)

  1. consider the new function g: A->B where B = f(A), which has the same graph as f
  2. g is bijective so it has an inverse g^-1: B->A
  3. the real valued inverse is the function h: B-> ℝ which has the same graph as g^-1
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27
Q

Parity: Odd function

A

Let f: A->ℝ for some A⊆ℝ

We say that f is an even function if, for all xЄA we have -xЄA and f(-x) = f(x)

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28
Q

Parity: even function

A

Let f: A->ℝ for some A⊆ℝ

We say that f is an odd function if, for all xЄA we have -xЄA and f(-x) = -f(x)

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29
Q

Symmetry of an even function

A

symmetrical about axis

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30
Q

Symmetry of odd function

A

Symmetrical about a point

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31
Q

Periodicity

A

Let f:A->B for some A⊆ℝ we say that f is periodic if there exists a number ω>0 such that, for all xЄA we have (x+ω)ЄA and f(x+ω) = f(x)

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32
Q

fundamental period

A

if f is periodic and has a minimum period ω>0, then this ω is the fundamental period

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33
Q

Monotonicity: increasing

A

Let f: A->ℝ for some A⊆ℝ.
we say that f is increasing if for all x,x’ЄA:
x <= x’ => f(x) <= f(x’)

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34
Q

Monotonicity: strictly increasing

A

Let f: A->ℝ for some A⊆ℝ.
we say that f is strictly increasing if for all x,x’ЄA:
x <= x’ => f(x) < f(x’)

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35
Q

Monotonicity: decreasing

A

Let f: A->ℝ for some A⊆ℝ.
we say that f is decreasing if for all x,x’ЄA:
x < x’ => f(x) >= f(x’)

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36
Q

Monotonicity: strictly decreasing

A

Let f: A->ℝ for some A⊆ℝ.
we say that f is strictly decreasing if for all x,x’ЄA:
x < x’ => f(x) > f(x’)

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37
Q

strictly increasing/ decreasing function are…

A

invertible

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38
Q

how to make a periodic function invertible

A

restrict the domain to make the function injective

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39
Q

Boundedness: Upper bound

A

Let A⊆ℝ.

we say that bЄℝ is an upper bound of A if x<=b for all xЄℝ

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40
Q

Boundedness: lower bound

A

Let A⊆ℝ.

we say that bЄℝ is an lower bound of A if x>=b for all xЄℝ

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41
Q

Boundedness: bounded above

A

Let A⊆ℝ.

A has an upperbound

42
Q

Boundedness: bounded below

A

Let A⊆ℝ.

A has a lowerbound

43
Q

boundedness: bounded

A

Let A⊆ℝ.

A is both bounded above and bounded below.

44
Q

how many lower/upper bounds

A

infinitely many

45
Q

Supremum (supA)

A

if A is bounded above, then the minimum of the upperbounds of A is called the supremum

46
Q

Infimum (infA)

A

If A is bounded below, then the maximum of the lowerbounds of A is called the infimum

47
Q

maximum (maxA)

A

largest defined element of A

48
Q

minimum (minA)

A

smallest defined element of A

49
Q

if A is unbounded above what is the supremum and maximum

A

supA = ∞

no maximum

50
Q

if A in unbounded below what is the infimum and minimum

A

infA = -∞

no minimum

51
Q

extended real line

A

all real numbers inc infinity and minus infinity

52
Q

bounded/unbounded above functions

A

we say that f is (un)bounded above if the range f(A) is (un)bounded above

53
Q

bounded/unbounded below functions

A

we say that f is (un)bounded above if the range is (un)bounded below

54
Q

tending to infinity definition

A

Let f:A->ℝ, A⊆ℝ ( Assuming unbounded above)

we say that the lim x->∞ f(x) = ∞ if for all M>0 there exists ℕЄℝ such that for all xЄA if x>N then f(x)>M

55
Q

tending to - infinity definition

A

Let f:A->ℝ, A⊆ℝ ( Assuming unbounded above)

we say that the lim x->∞ f(x) = -∞ if for all M>0 there exists ℕЄℝ such that for all xЄA if x>N then f(x)>-M

56
Q

tending to a point definition

A

Let f:A->ℝ, A⊆ℝ ( Assuming unbounded above)
we say that the lim x->∞ f(x) = l (where l⊆ℝ) if for all Ɛ>0 there exists NЄℝ such that, for all xЄℝ if x>N then |f(x)-l|

57
Q

when proving tending functions.

A

work out an expression for M/Ɛ then use the definition forwards

58
Q

Accumulation point

A

for all values of delta there exists a value of x in the set a where |x-a| lies between 0 and delta

59
Q

function f:A->B

A

let A and B be sets. A function from A to B is a rule that associates every element of A to exactly one element of B.

60
Q

function tends to infinity as x tends to a definition

A

let f:A->ℝ, A⊆ℝ, aЄℝ (assume that a is an acculmulation point of A) we say that lim x->a of f(x)=∞ if for all M>0 there exists 𝛿>0 such that for all xЄA: if 0M

61
Q

function tends to minus infinity as x tends to a definition

A

let f:A->ℝ, A⊆ℝ, aЄℝ (assume that a is an acculmulation point of A) we say that lim x->a of f(x)=-∞ if for all M>0 there exists 𝛿>0 such that for all xЄA: if 0

62
Q

function tends to a limit l as x tends to a definition

A

let f:A->ℝ, A⊆ℝ, aЄℝ (assume that a is an acculmulation point of A) we say that lim x->a of f(x)=l if for all Ɛ>0 there exists 𝛿>0 such that for all xЄA: if 0

63
Q

what does writing a limit assume

A

the existence and uniqueness of the limit

64
Q

uniqueness of limits proposition

A

let f:A->ℝ, A⊆ℝ, aЄℝ (assume that a is an acculmulation point of A) Let l1,l2ЄℝU{∞,-∞}
if lim x->a f(x) = l1 an d lim x->a f(x) = l2
then l1 = l2

65
Q

Right Limit

A

lim x->a+ f(x) = lim x->a f| A∩(a,∞)(x)

66
Q

Left Limit

A

lim x->a- f(x) = lim x->a f| A∩(-∞,a)(x)

67
Q

Limit based on left an right limits

A

let f:A->ℝ, A⊆ℝ, aЄℝ (assume that a is an acculmulation point of A)
The the limit lim x->a f(x) exists if and only if both one sided limits lim x->a+ f(x) and lim x->a- f(x) exist and are equal

68
Q

Algebra of Limits

A

let f,g:A->ℝ, A⊆ℝ, aЄℝ (assume that a is an acculmulation point of A)
Assume that lim x->a f(x) = l and lim x->a g(x) = m for some l,mЄℝ
then:
1. lim x->a (f(x)+g(x)) = l + m
2. lim x->a (f(x).g(x)) = lm

69
Q

sandwhich theorem

A

f(x)<=g(x)<=h(x)

if lim x->a f(x) = lim x->a h(x) = l then lim x->a g(x) = l

70
Q

Comparison Theorem

A

f(x)<= g(x)

if lim x->a f(x) = l and lim x->a g(x) = m for some l,mЄℝ. The l<=m

71
Q

continuous function definition

A

let f:A->ℝ, A⊆ℝ.

we say that the function if continuous if f is continuous as all point aЄA.

72
Q

continuous at a definition

A

let f:A->ℝ, A⊆ℝ.

Let aЄA. The function f is continuous at a if for all Ɛ>0 there exists 𝛿>0 such that for all xЄA: if |x-a|

73
Q

in general which functions are continuous

A
  1. polynomials and rational functions
  2. modulus functions
  3. nth root function
  4. Trigonometric functions (+inverses)
  5. exponential and logarithm functions
74
Q

accumulation point/ continuity of functions proposition

A

let f:A->ℝ, A⊆ℝ, Let aЄA.

(i) if a is an acculmulation point of A, then f is continuous at a.
(ii) if a is an acculmulation point of A, then: f is continuous at a <=> lim x->a f(x) = f(a)

75
Q

if a is an acculmulation point of A, then: f is continuous at a <=>

A

<=> lim x->a f(x) = f(a)

76
Q

ALGEBRA OF CONTINUOUS FUNCTIONS

A

let f,g:A->ℝ, A⊆ℝ, Let aЄA and assume f and g are continuous at a. Then:

(i) The sum f+g is continuous at a
(ii) The product f.g is continuous at a
(iii) if f(a) != 0, then 1/f is continuous at a

77
Q

composition of continuous functions

A

let f:A->ℝ g:B->ℝ A,B⊆ℝ such that f(A)⊆B. Let aЄℝ and assume that lim x->a f(x) = b for some bЄR where g is continuous at b. Then:
lim x->a g(f(x)) = g(b)

78
Q

if both f and g are continuous then

A

the composition of g and f is continuous

79
Q

tip for finding a limit

A

rationalise

80
Q

lim x->0 sinx/x =

A

= 1

81
Q

lim x->0 1-cosx/x^2 =

A

= 1/2

82
Q

lim x->∞ (1+1/x)^2 =

A

= e

83
Q

lim x-> -∞ (1+1/x)^2 =

A

= e

84
Q

lim x->0 log(1+x)/x =

A

= 1

85
Q

lim x-> 0 e^x -1 / x =

A

= 1

86
Q

lim x->∞ e^x/x^b

A

= ∞

87
Q

lim x->∞ logx/x^b =

A

= 0

88
Q

lim x->0+ x^blogx =

A

= 0

89
Q

A^B =

A

exp(BlogA)

90
Q

Itermediate Value Theorem

A

Let a,bЄℝ. Let f:[a,b]->ℝ continuous. Let cЄℝ such that:
f(a)<=c <= f(b)
Then there exists xЄ[a,b] such that f(x)=c

91
Q

Boundedness Theorem

A

Let a,bЄℝ aℝ be continuous. Then f is bounded and attains its bounds. In other words, there exists x1,x2Є[a,b] such that f(x1)<= f(x) <= f(x2) for all xЄ[a,b]

92
Q

Differentiability at a definition

A

Let f:A->ℝ for some A⊆ℝ.
Let aЄA. if the limit
lim x->a f(x)-f(a)/x-a = l for some lЄℝ.
Then we say that f is differentiable at a. in this case the limit l is called the derviative and is denoted by f’(a)

93
Q

differentiable definiton

A

we say that f:A->ℝ is differentiable if f is differentiable at every point aЄℝ

94
Q

Differentiability and continuity

A

Let f:A->ℝ for some A⊆ℝ. let aЄℝ. if f is differentiable at point a, then f is continuous at a.

95
Q

for a function f: A->ℝ A⊆ℝ, let aЄℝ. What statements are equivilent?

A
  1. f is differentiable at a
  2. There exists a function g: A->ℝ which is continuous at a such that
    f(x) = f(a) + g(x).(x-a)
    moreover, in this case g(a) = f’(a)
96
Q

Algebra of differentiable functions

A
  1. f+g is differentiable at a and (f+g)’(a) = f’(a) + g’(a)
  2. f.g is differentiable at a and (f.g)’(a) = f’(a)g(a) + f(a)g’(a)
  3. f/g is differentiable at a and (f/g)’(a) = f’(a)g(a) - f(a)g(a) / (g(a))^2
97
Q

product rule

A

(f.g)’(a) = f’(a)g(a) + f(a)g’(a)

98
Q

quotient rule

A

(f/g)’(a) = f’(a)g(a) - f(a)g(a) / (g(a))^2

99
Q

g’(f(a)) =

A

1/f(a)

100
Q

higher order derivatives at a point definition

A

we say that f is n times differentiable at a point a if a is an element of the domain of f(a)

101
Q

n times continuously differentiable definition

A

if f is n times differentiable and continuous

102
Q

Any function that is N times differentiable is always…

A

n-1 times continuously differentiable

1st year Maths (24 decks)

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