Flashcards
C1: Definitions
Cards in this deck (76)
- 1A map/mappingA map, or mapping, f from a set A into a set B is a rule that assigns to each element x ∈ A a unique element f(x) ∈ B, denoted by
f : A → B, or A ∋ x → f(x) ∈ B. - 2Domain & codomain of a functionWe call the set A = dom(f) the domain (of definition) of f and the set B the
codomain (or target) - 3Range of a functionGiven a mapping f : A → B, the range of f, denoted by rng(f) ⊂
B, is the set of all image elements, or the image of the domain A. Thus
rng(f) := f(A) = {f(x) |x ∈ A = dom(f)} ⊆ B
We say f maps A onto rng(f) - 4Largest Integer part functionThe largest integer part function is a map
f : R → Z defined by the rule
R ∋ x → f(x) = [x], where n = [x] ∈ Z is the largest integer n ≤ x. - 5A sequenceA sequence is a map s that assigns to each natural number a real number
s: N → R, where n → s(n) = sn - 6A functionA function f from a set A into a set B ⊂ R is a rule that assigns to each element x ∈ A a unique element y ∈ B. The number f(x) ∈ B ⊂ R
- 7Image & value of a functionThe number f(x) ∈ B ⊂ R , the image of x under f, is also called the value of f at x.
- 8Independent variableWe shall focus on functions f : A ⊂ R → B of a single real variable, where B ⊆
R. Here x, the input to f, is called the independent variable - 9Dependent variabley = f(x), the unique output, is called the dependent variable, since it depends on x.
- 10The Identity functionThe function IdA : A → A given by the rule IdA(x) = x,∀x ∈ A, is called the
identity function. - 11The Zero functionThe function ZA : A → R defined by the rule ZA(x) = 0,∀x ∈
A, is called the zero function. - 12The Constant Unit FunctionThe function UA : A → R defined by the rule UA(x) = 1,∀x ∈ A, is called the constant unit function.
- 13The Natural Domain of a functionThe natural domain of a function f is the set of all real values of x for which the formal rule makes sense, and gives a unique real value
dom(f) := {x|f(x) is defined}
Unless otherwise specified, we shall take the domain of a function to be its natural domain - 14The Natural Range of a functionThe natural range is the corresponding set of values f(x).
Unless otherwise specified, we shall take the target B = R. Thus is we assume that f : dom(f) → R - 15GraphLet f : A ⊂ R → R be a function. Then the graph of f is the set of points (x,f(x)) in the plane R2, for which x is in the domain of f. Thus
graph(f): = A × f(A) =
{(x;y) ∈ R2 |x ∈ A,y = f(x)}
⊂ R2 - 16Root (graph property)If f(c) = 0 then c is called a root of f and the point (c,0) is where the graph(f) intercepts the x-axis.
- 17y-interceptIf f(0) = b, then the point (0,b) is where the graph(f) intercepts the y-axis
- 18Asymptotes (horizontal/vertical)A graph which is increasingly close to a vertical (horizontal) line x = a (y = b)
has a vertical (horizontal) asymptote at x = a (y = b). - 19Positive/negative on f/IIf f(x) > 0 (< 0) for all x ∈ I, we say that f is positive (negative) on I.
- 20Increasing/decreasing on f/IIf f(b) > f(a) (< f(a)) for all b > a ∈ I, we say that f is increasing
(decreasing) on I. - 21The Vertical Line TestA function has exactly one output. Thus a graph represents a function if and only if every vertical line intersects the graph in at most one point.
- 22InjectiveA function f : A → B is said to be injective (or one-to-one) if for all (∀) x1,x2 ∈ A, then x1 = x2 ⇒ f(x1) = f(x2) or f(x1) = f(x2) ⇒ x1 = x2
- 23SurjectiveA function f : A → B is said to be surjective (or onto) if rng(f) =
B. That is for all (∀) y ∈ B, there exists (∃) x ∈ A such that y = f(x). - 24BijectiveA function f : A → B is bijective, if it is injective and surjective.
- 25The Horizontal Line TestA function f : A → B is injective if and only if any horizontal line drawn through any y ∈ B intersects the graph of f at most once and surjective if it intersects the graph of f at least once.
- 26The Restricted FunctionGiven f : A → B, we can define g : C → B by x → g(x) = f(x)
for all x ∈ C ⊂ A = dom(f), where g = f|C is called the restriction of f to C. - 27An Even FunctionLet f : A → B then f is called
even if f satisfies f(−x) = f(x) for all x ∈ A
Note if f is odd or even, then if x ∈ dom(f) we must have −x ∈ dom(f). - 28An Odd FunctionLet f : A → B then f is called
odd if f satisfies f(−x) = −f(x) for all x ∈ A
Note if f is odd or even, then if x ∈ dom(f) we must have −x ∈ dom(f). - 29Power FunctionsGiven k = m/n is rational, then f : A ⊂ R → R given by
f(x) = x^k where y = x^(m/n) ⇔ y^n = x^m = (1/x)^−m
is called a power function. Here k is the power or exponent and x the base. - 30PolynomialsA polynomial of degree n is the function P : R → R given by
P(x) = anxn + an−1xn−1 + ··· + a1x + a0 =
n
∑
k=0
akxk
where the coefficients ak ∈ R and the leading coefficient an = 0. - 31Fundamental Theorem of Algebra & FactorizationValues of x satisfying P(x) = 0 are known as the roots
of the polynomial. By the fundamental theorem of algebra, if P(x) has r ≤ n roots xk satisfying P(xk) = 0 for k = 1,...,r, then we can factorize the P(x) by writing
P(x) = g(x) ×
r
∏
k=1
(x − xk)
and g, a polynomial of degree n−r, is a product of irreducible quadratic factors. - 32Linear FunctionThe graph of the linear function f : R → R, a polynomial of degree 1, defined by x → f(x) ≡ mx + c = m(x − r), m = 0 is a straight line, of inclination α = tan−1(m) to the positive x-axis.
- 33A Quadratic FunctionA quadratic function f : R → R is a polynomial of degree 2. A quadratic may have two roots, a single repeated root, or no roots. We can write
x → f(x) = ax2 + bx + c =
a(x − α)(x − β)
a(x − α)2
a
[
(x − α)2 + β2] , a = 0 - 34A Rational FunctionA rational function f : A ⊂ R → R is given by the rule
R(x) =
P(x) Q(x)
=
anxn + ··· + a1x + a0 bmxm + ··· + b1x + b0
where P(x) and Q(x) are polynomials. The natural domain of R(x) is {x|Q(x) = 0}. - 35An Infinite SequenceWe recall that an infinite sequence (ak)k≥0 is a mapping an : N → R defined by n → an.
- 36nth Partial sum of an infinite sequenceGiven a fixed sequence (ak)k≥0 then for each x ∈ R we can define the nth partial sum to be the function x → Pn(x) definite by the finite sum
Pn(x) =
n
∑
k=0
anxk = a0 + a1x + ··· + anxn - 37Power SeriesGiven a fixed sequence (ak)k≥0, then we define the power series
to be the function f : R → R defined by
f(x) =
∞
∑
k=0
anxk = lim
n→∞
n
∑
k=0
anxk for |x| < R - 38Radius of Convergence (linked to previous definition)In particular f(0) = a0. Here dom(f) = (−R,R) are those values for which the infinite series converges, where R ≥ 0 is known as the radius of convergence of the power series.
- 39Piece-wise functionsPiecewise defined functions are functions that are defined by different rules on different parts of their domains.
- 40Modulus FunctionThe modulus function |x| is defined as the distance from x to 0. We can define f(x) = |x| equivalently as a piecewise defined function:
|·| : R → R, x → f(x) = |x| =
{x for x ≥ 0
−x for x < 0 - 41Heaviside FunctionThe Heaviside function H, is defined by
H: R → R, x → H(x) =
{
1 for x ≥ 0 0 for x < 0 - 42Signum FunctionThe sign, or signum, function is defined by
sgn : R → R, x → sgn(x) =
1 for x > 0
0 for x = 0
−1 for x < 0 - 43Largest Integer part FunctionThe (largest) integer part function is defined by
[·] : R → Z, x → [x] = n ≤ x
where n is the largest integer less than or equal to x. It graph
E x
T
y = [x] - 44The Characteristic Indicator FunctionThe characteristic, or indicator, function of a subset S ⊂ R is defined by
χS : R → R, x → χS(x) =
{
1 for x ∈ S
0 for x / ∈ S - 45Dirichlet's FunctionThe (bizarre) Dirichlet function is defined by hD : R → Q, x → h(x) =
{ 1
n if x = m
n ∈ Q
0 if x /
∈ Q
so that hD(1
2 ) = 1
2 , hD(1
3 ) = hD(2
3 ) = 1
3 , hD(1
4 ) = hD(3
4 ) = 1
4 , etc. - 46Scalar Multiplication function (CF1)Let f : A → B and g: A′ → B′ be two functions •
The scalar multiple function kf is the function kf : A → R given by the
rule (kf)(x): = kf(x), for any k ∈ R - 47The Sum of Functions (CF 2)Let f : A → B and g: A′ → B′ be two functions
The sum of functions f + g is the function f + g: A ∩ A′ → R given by the rule (f + g)(x): = f(x) + g(x). - 48The Difference of Functions (CF3)Let f : A → B and g: A′ → B′ be two functions
The difference of functions f − g is the function f − g: A ∩ A′ → R given by the rule (f − g)(x): = f(x) − g(x). - 49Product of Functions (CF4)Let f : A → B and g: A′ → B′ be two functions
The product of functions is the function fg: A ∩ A′ → R given by the rule (fg)(x): = f(x)g(x). - 50Quotient of Functions (CF5)Let f : A → B and g: A′ → B′ be two functions
The quotient of functions is the function f/g: dom(f/g) → R given by the rule (f/g)(x): = f(x)/g(x), where dom(f/g) := {A ∩ A′ |g(x) = 0}. - 51Additive identity and constant multiple property of all functionsFor any function f : A → B added the identity function ZA : A → R , defined by the rule ZA(x) = 0 ∀x ∈ A, or multiplied by the constant unit function UA : A →
R ∀x ∈ A, defined by the rule UA(x) = 1 ∀x ∈ A, we have
f + ZA = f, UAf = f - 52Equal FunctionsWe say that two functions f, g are equal, denoted by f = g, if their domains and targets are the same and
f(x) = g(x) for all x ∈ A = dom(f) or f(x) − g(x) = ZA(x) - 53Composite functionGiven functions f : dom(f) → B, g: dom(g) → C, then if rng (g)∩
dom(f) = ∅ we can define the composite function f ◦ g: dom(f ◦ g) → B, by
(f ◦ g)(x) := f(g(x))
(1.11)
where dom(f ◦ g) := {x ∈ dom(g)|g(x) ∈ dom(f)} ⊆ dom(g). - 54Inverse functionLet f and g be functions such that
(g ◦ f)(x) ≡ g (f(x)) = x for x ∈ A = dom(f) (f ◦ g)(x) ≡ f (g(x)) = x for x ∈ B = dom(g)
then g is called the inverse function of f, and we write g = f−1. Thus
f−1 ◦ f = IdA and f ◦ f−1 = IdB - 55Transcendental FunctionsFunctions that are not algebraic are called transcendental functions. One type of transcendental function is trigonometric functions.
- 56Periodicity of cosinecos(x + 2π) = cosx,
cos(x −π/2) = sinx, - 57Periodicity of Sinesin(x + 2π) = sinx
sin(x +π/2) = cosx - 58Additive formulaecos(x ± y) = cosxcosy ∓ sinxsiny
sin(x ± y) = sinxcosy ± cosxsiny
tan(x ± y) =tanx ± tany/ 1 ∓ tanxtany - 59Double Angle Formulaecos(2x) = cos^2 x − sin^2 x = 2cos^2 x − 1 = 1 − 2sin^2x
2sin2 x sin(2x) = 2sinxcosx - 60sineDomain: [−π/2,π/2]
Range: [−1,1] - 61CosineD: [0,π]
R: [−1,1] - 62TanD: (−π/2,π/2)
R: (−∞,∞) - 63CosecD:[−π/2,0) ∪ (0,π/2]
R: (−∞,−1] ∪ [1,∞) - 64SecD: [0,π/2) ∪ (π/2,π]
R: (−∞,−1] ∪ [1,∞) - 65CotD:(0,π)
R: (−∞,∞) - 66Inverse SineThe inverse sine function sin−1 x is defined by, if
x = siny for y ∈ [−π/2,π/2] ⇒ y = sin−1 x for x ∈ [−1,1] - 67Inverse CosineThe inverse cosine function cos−1 x is defined by, if
x = cosy for y ∈ [0,π] ⇒ y = cos−1 x for x ∈ [−1,1] - 68Inverse TanThe inverse tangent function tan−1 x is defined by, if
x = tany for y ∈ (−π/2,π/2) ⇒ y = tan−1 x for x ∈ (−∞,∞) - 69Inverse SecThe inverse secant function sec−1 x is defined by, if
x = secy for y ∈ [0,π/2)∪(π/2,π] ⇒ y = sec−1 x for x ∈ (−∞,−1]∪[1,∞) - 70Inverse Cosecx = cosecy for y ∈ [−π/2,0) ∪ (0,π/2] ⇒ y = cosec−1 x for x ∈(−∞,−1] ∪ [1,∞)
- 71Inverse Cotx = coty for y ∈ (0,π) ⇒ y = cot−1 x for x ∈ (−∞,∞)
- 72Exponential FunctionFor any real positive number a > 0 an exponential function is a function f : R → R given by the rule
x → f(x) = a^x for any x ∈ R - 73ethe best definition for e is given using limits
e = lim
h→0
(1 + h)^1/h = lim
n→∞
(1 +1/n)^n
=
∞
∑
k=1
1 k!
= 2.7182818... - 74The Logarithmic FunctionThe inverse of the exponential function ax, where the base a >
0,a = 1 is the logarithmic function loga x, or
x = ay ⇒ y = loga x for x ∈ (0,∞) (1.19) In particular the inverse to ex is the natural logarithmic function lnx, or x = ey ⇒ y = lnx for x ∈ (0,∞) - 75Hyperbolic Functionshe hyperbolic functions cosh x and sinh x are defined by
coshx =
1 /2
( e^x + e−x)
, sinhx =
1 2
( ex − e−x) - 76Hyperbolic to Trigonometric Relationcoshx = cos(ix),
sinhx = −isin(ix)
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- Calculus (Chapter 1: Functions & Mappings) Flashcards on C1: Definitions, created by Luke Danaher on 22/12/2016.
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