RAC functions Flashcards
function
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.
set of inputs
domain
set of outputs
codomain
image/range
for every element from A, xЄA then the element of B associated to x by f is called the image
difference between codomain and image/range
codomain are the possible outputs
range/image are the actual outputs
image definition (function f: A->B)
the subset of B whose elements are the images of elements A. denoted by F(A)
f: A->B g: C->D f=g if…
- A=C
- B=D
- f(x) = g(x) for all xЄA
graph of symbol
Γ
image letters
f(A) = {f(x): xЄA}
graph letters
Γf = {(x,y)ЄAxB: y=f(x)}
how to tell if a graph is a graph of a function
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)
constructing function via restriction definintion
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
what is restriction
taking a smaller domain to only show part of the function
constructing a function via composition definition
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
indentity function definition
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
what happens if you do composition with an identity function
nothing
Constant Function definition
A function f:A->B is said to be constant if there exists bЄB such that f(x)=b for all xЄA
what do functions have to be to be invertible
one-to-one
invertibility definition
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
Injective (one-to-one)
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.
surjective
f(A) = B
if every image point has at least one preimage in A
bijective
both surjective and injective.
if every image point has exactly one preimage in A
g: B->A is the inverse of f:A->B iff
g∘f = idₐ and f∘g = idb
real valued function of a real variable
f: A->ℝ , where A⊆ℝ
domain convention
- assume codomain is ℝ
- the domain is the largest subset of ℝ where the given subset makes sense
finding the real valued inverse
(the function f: A-> B is injective)
- consider the new function g: A->B where B = f(A), which has the same graph as f
- g is bijective so it has an inverse g^-1: B->A
- the real valued inverse is the function h: B-> ℝ which has the same graph as g^-1
Parity: Odd function
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)
Parity: even function
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)
Symmetry of an even function
symmetrical about axis
Symmetry of odd function
Symmetrical about a point
Periodicity
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)
fundamental period
if f is periodic and has a minimum period ω>0, then this ω is the fundamental period
Monotonicity: increasing
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’)
Monotonicity: strictly increasing
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’)
Monotonicity: decreasing
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’)
Monotonicity: strictly decreasing
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’)
strictly increasing/ decreasing function are…
invertible
how to make a periodic function invertible
restrict the domain to make the function injective
Boundedness: Upper bound
Let A⊆ℝ.
we say that bЄℝ is an upper bound of A if x<=b for all xЄℝ
Boundedness: lower bound
Let A⊆ℝ.
we say that bЄℝ is an lower bound of A if x>=b for all xЄℝ
Boundedness: bounded above
Let A⊆ℝ.
A has an upperbound
Boundedness: bounded below
Let A⊆ℝ.
A has a lowerbound
boundedness: bounded
Let A⊆ℝ.
A is both bounded above and bounded below.
how many lower/upper bounds
infinitely many
Supremum (supA)
if A is bounded above, then the minimum of the upperbounds of A is called the supremum
Infimum (infA)
If A is bounded below, then the maximum of the lowerbounds of A is called the infimum
maximum (maxA)
largest defined element of A
minimum (minA)
smallest defined element of A
if A is unbounded above what is the supremum and maximum
supA = ∞
no maximum
if A in unbounded below what is the infimum and minimum
infA = -∞
no minimum
extended real line
all real numbers inc infinity and minus infinity
bounded/unbounded above functions
we say that f is (un)bounded above if the range f(A) is (un)bounded above
bounded/unbounded below functions
we say that f is (un)bounded above if the range is (un)bounded below
tending to infinity definition
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
tending to - infinity definition
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
tending to a point definition
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|
when proving tending functions.
work out an expression for M/Ɛ then use the definition forwards
Accumulation point
for all values of delta there exists a value of x in the set a where |x-a| lies between 0 and delta
function f:A->B
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.
function tends to infinity as x tends to a definition
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
function tends to minus infinity as x tends to a definition
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
function tends to a limit l as x tends to a definition
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
what does writing a limit assume
the existence and uniqueness of the limit
uniqueness of limits proposition
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
Right Limit
lim x->a+ f(x) = lim x->a f| A∩(a,∞)(x)
Left Limit
lim x->a- f(x) = lim x->a f| A∩(-∞,a)(x)
Limit based on left an right limits
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
Algebra of Limits
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
sandwhich theorem
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
Comparison Theorem
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
continuous function definition
let f:A->ℝ, A⊆ℝ.
we say that the function if continuous if f is continuous as all point aЄA.
continuous at a definition
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|
in general which functions are continuous
- polynomials and rational functions
- modulus functions
- nth root function
- Trigonometric functions (+inverses)
- exponential and logarithm functions
accumulation point/ continuity of functions proposition
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)
if a is an acculmulation point of A, then: f is continuous at a <=>
<=> lim x->a f(x) = f(a)
ALGEBRA OF CONTINUOUS FUNCTIONS
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
composition of continuous functions
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)
if both f and g are continuous then
the composition of g and f is continuous
tip for finding a limit
rationalise
lim x->0 sinx/x =
= 1
lim x->0 1-cosx/x^2 =
= 1/2
lim x->∞ (1+1/x)^2 =
= e
lim x-> -∞ (1+1/x)^2 =
= e
lim x->0 log(1+x)/x =
= 1
lim x-> 0 e^x -1 / x =
= 1
lim x->∞ e^x/x^b
= ∞
lim x->∞ logx/x^b =
= 0
lim x->0+ x^blogx =
= 0
A^B =
exp(BlogA)
Itermediate Value Theorem
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
Boundedness Theorem
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]
Differentiability at a definition
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)
differentiable definiton
we say that f:A->ℝ is differentiable if f is differentiable at every point aЄℝ
Differentiability and continuity
Let f:A->ℝ for some A⊆ℝ. let aЄℝ. if f is differentiable at point a, then f is continuous at a.
for a function f: A->ℝ A⊆ℝ, let aЄℝ. What statements are equivilent?
- f is differentiable at a
- 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)
Algebra of differentiable functions
- f+g is differentiable at a and (f+g)’(a) = f’(a) + g’(a)
- f.g is differentiable at a and (f.g)’(a) = f’(a)g(a) + f(a)g’(a)
- f/g is differentiable at a and (f/g)’(a) = f’(a)g(a) - f(a)g(a) / (g(a))^2
product rule
(f.g)’(a) = f’(a)g(a) + f(a)g’(a)
quotient rule
(f/g)’(a) = f’(a)g(a) - f(a)g(a) / (g(a))^2
g’(f(a)) =
1/f(a)
higher order derivatives at a point definition
we say that f is n times differentiable at a point a if a is an element of the domain of f(a)
n times continuously differentiable definition
if f is n times differentiable and continuous
Any function that is N times differentiable is always…
n-1 times continuously differentiable
