functions definitions 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
image
if f: A->B and x∈A, then the element of B associated to x by f is called the image(range) of x via f. denoted by f(x).
pre image
if f: A->B and y∈B then any element x∈A such that f(x)=y is called the pre image of y via f.
range
The image(range) of a function f:A->B is the subset of B whose elements are the images of the elements of A.
Graph of a function
Let f: A->B. The graph of f is the set ⌈f = {(x,y)∈AxB: y=f(x)}
Restriction
Let f:A->B. Let S⊆A. The restriction of f to s if the function f|ₛ: S->B defined by f|ₛ(x)= f(x) for all x∈S
Composition
Let f: A->B and g: C->D. 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 x∈A
identity function
let A be a set.The identity function of A is the function id: A->A defined by id(x) = x for all x∈A
constant function
a function f:A->B is said to be constant if there exists b∈B such that f(x)=b for all x∈A.
invertibility
Let f:A-B. we say that f is invertible if there exists a function g:B->A such that for all x∈A and y∈B f(x) = u <=> g(y) = x
injective
Let f:A->B we say that f is injective (or one-to-one) if for all x, x’∈A: x!=x’ => f(x)!=f(x’).
or
f is injective iff every y∈B has at most one preimage in A
srujective
Let f:A->B we say that f is surjective (or onto) if f(A)=B
or
f is surjective iff every y∈B has at least one preimage in A
injective simply
Two inputs don’t give the same output
surjective simply
An input can give any output even if its a repeated output
bijective
|Let f:A-B why say that f is bijective if it is both injective and surjective i.e. it has exactly one preimage in A for every y∈B
parity
Let f: A->ℝ for some A⊆ℝ. we say that:
(1) f is an even function if for all x∈A we have -x∈A and f(-x) = f(x)
(2) f is an odd function if for all x∈A we have -x∈A and f(-x) = -f(x)
real valued function of a real variable
f:A->ℝ, where A⊆ℝ
periodicity
let f:A->ℝ for some A⊆ℝ we say that f is periodic if there exists a number ω>0 such that for all x∈A we have x+ω∈A and f(x+ω)=f(x)
fundamental period
if f is periodic and has a minimum period ω>0, then this ω is called the fundamental period of f
monotonicity
Let f:A->ℝ for some A⊆ℝ. we say that:
1) f is increasing for all x,x’∈A x≤x’ => f(x)≤f(x’
(2) f is strictly increasing for all x,x’∈A x f(x) f(x)f(x’)
Boundedness
Let A⊆ℝ. We say that:
(1) b∈ℝ is an upperbound of A is x≤b for all x∈A
(2) b∈ℝ is an lowerbound of A is x≥b for all x∈A
(3) A is bounded above if it has an upperbound
(4) A is bounded below if it has a lowerbound
(5) A is bounded if it is both bounded above and below
supremum
If A is bounded above, then the minimum of the upperbounds of A is called the supremum of A
Infimum
If A is bounded below, then the maximum of the lowerbounds of A is called the infimum of A
inf(A) = ∞
if A is unbounded below then we write inf(A) = ∞
sup(A) = ∞
if A is unbounded above then we write sup(A) = ∞
boundedness of a function defined by the range
Let f:A->ℝ, A⊆ℝ. Then:
(i) we say that f is (un)bounded above if the range f(A) is (un)bounded above
(ii) we say that g is (un)bounded below if the range f(A) is (un)bounded below
(iii) sup(f) = sup(f(A)) and inf(f) = inf(f(A))
(iv) if they exist then max(f) = max(f(A)) and min(f) = min(f(A))
min
min(A) is the smallest element of A
max
max(A) is the largest element of A
limit as x tends to infinity of a function is infinity
Let f:A->ℝ, A⊆ℝ. we say that lim x->∞ f(x) = ∞ is for all m>0 there exists N∈ℝ such that for all x∈A if x>N then f(x)>M
limit as x tends to infinity of a function is minus infinity
Let f:A->ℝ, A⊆ℝ. we say that lim x->∞ f(x) = -∞ is for all m>0 there exists N∈ℝ such that for all x∈A if x>N then f(x) is less than -M
limit as x tend to infinity of a function is a point l
Let f:A->ℝ, A⊆ℝ. we say that lim x->∞ f(x) = l is for all ε>0 there exists N∈ℝ such that for all x∈A if x>N then |f(x)-l| is less than epsilon
accumulation point
Let A⊆ℝ and a∈ℝ. a is said to be an acculmulation point of A if for all 𝛿>0 there exists x∈A: 0
limit as x->a of a function is infinity (a is an accumulation point of A)
Let f:A->ℝ, A⊆ℝ, Let a∈ℝ. we say that lim x->a f(x) = ∞ is for all m>0 there exists 𝛿>0 such that for all x∈A 0 if is less than |x-a| is less than delta then f(x) is greater than M
limit as x->a of a function is minus infinity (a is an accumulation point of A)
Let f:A->ℝ, A⊆ℝ, Let a∈ℝ. we say that lim x->a f(x) = -∞ is for all m>0 there exists 𝛿>0 such that for all x∈A if 0 is less than |x-a| is less than delta then f(x) is less than -M
limit as x->a of a function is a point l (a is an accumulation point of A)
Let f:A->ℝ, A⊆ℝ, Let a∈ℝ, Let l∈ℝ. we say that lim x->a f(x) = l is for all ε>0 there exists 𝛿>0 such that for all x∈A if 0 is less than |x-a| is less than delta then |f(x)-l| is less than epsilon
acculmulation point simply
A point in the interval which is in the set
continuous
Let f:A->ℝ, A⊆ℝ
(i) let a∈ℝ. The function f is continuous at a if for all ε>0 there exists 𝛿>0 such that for all x if |x-a| is less than delta then |f(x)-f(a)| is less than epsilon
(ii) we say that the function f is continuous at all points a∈A
Differentiability
let f: Ω->ℝ for some Ω⊆ℝ
(i) let a∈ℝ if the limit
limₓ₋>ₐ (f(x)-f(a))/(x-a) = l for some l∈ℝ
then we say that f is differentiable at a.
in this case the limit l is called the derivative of f at a and is denoted by f’(a)
(ii) we say that f:Ω->ℝ is differentiable if f is differentiable at every point a∈ℝ
(iii) more generally, if A⊆Ω, we say that f is differentiable on A if f is differentiable at all points a∈A
higher order derivatives
let f:Ω->ℝ for some Ω⊆ℝ. let n∈ℕ. we say that f is:
(i) n times differentiable at a point a∈Ω if a is an element of the domain of f(a)
(ii) n times differentiable if f is n times differentiable if f is n time differentiable at every a∈Ω
(iii) we say that f is n times continuously differentiable if f is n times differentiable and f(a) is continuous
(iv) we say that f is infintely differentiable if f is n times differentiable for all natural numbers n
global maxima
Let f:Ω->ℝ for some Ω⊆ℝ. A point a∈Ω is said to be a global maximum for for f if f(a)>=f(x) for all x∈Ω
local maxima
Let f:Ω->ℝ for some Ω⊆ℝ. A point a∈Ω is said to be a local maximum point for f if there exists r>0 such that f(a)>= for all x∈Ω∩(a-r, a+r)
global minima
Let f:Ω->ℝ for some Ω⊆ℝ. A point a∈Ω is said to be a global minimum point for f if f(a)<=f(x) for all x∈Ω
local minima
Let f:Ω->ℝ for some Ω⊆ℝ. A point a∈Ω is said to be a local minimum point for f if f(a)<=f(x) for all x∈Ω∩(a-r,a+r)
stationary point
a is a stationary point (or critical point) for f if f is differentiable at a and f’(a)=0
interior point
Let Ω⊆ℝ where a point a∈ℝ is called an interior point of Ω if there exists r>0 such that (a-r,a+r)⊆Ω
Convexity
let f:I->ℝ, where I is an interval, we say that:
(i) f is a convex function if f((1-t)a+tb)<= (1-t)f(a) + tf(b) for all a,b∈I and t∈[0,1]
(ii) more generally, we say that f:Ω->ℝ, Ω⊆ℝ, is convex on an interval I⊆ℝ if f|ᵢ is convex
Concavity`
let f:I->ℝ, where I is an interval, we say that:
(i) f is a concave function if f((1-t)a+tb)>= (1-t)f(a) + tf(b) for all a,b∈I and t∈[0,1]
(ii) more generally, we say that f:Ω->ℝ, Ω⊆ℝ, is concave on an interval I⊆ℝ if f|ᵢ is concave
Vertical Asymtote
the line x=a is a vertical asymtote of the graph of the function f if
limₓ₋>ₐ₊f(x) = ±∞ or limₓ₋>ₐ₋f(x) = ±∞
Horizontal Asymtote
The line y=b is a horizontal asymtote of the graph f if
limₓ₋>∞f(x) = b or limₓ₋>-∞f(x) = b
Oblique Asymtote
Let m,q∈ℝ where m!=0. we say that the line (y=mx+q) is called an oblique asymtote for the graph f if limₓ₋>∞(f(x)-(mx+q)) = 0 or limₓ₋>-∞(f(x)-(mx+q)) = 0
tangent line
The tangent line to the curve at P is the limit of the line Lpq as Q->P along the curve
normal line
The normal line to the curve at P is the line through P which is perpendicular to the tangent at P
Parametricisation
given two real-valued functions f,g:Ω->ℝ of a real variable, we can describe a curve as the set of points (x,y)∈ℝ of the form
