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REAL FUNCTION

A function is a law that associates an element of one set with only one element of a second set

f: A → B, a → b = f(a)

A real function of a real variable is a function f: X ⊆ ℝ → ℝ, x → y = f(x)

f: ℝ → ℝ, x → f(x) = x2

X domain of f; f(X) = {y ∈ ℝ, y = f(x)} image of f; G(f) = {(x,y) ∈ ℝ2, x ∈ X, y = f(x)} ⊆ ℝ2 graph of f

f: X → ℝ, g: X → ℝ; f+g: X → ℝ, x → (f+g)(x) = f(x)+g(x); f-g: X → ℝ, x → (f-g)(x) = f(x)-g(x); f⋅g: X → ℝ, x → (f⋅g)(x) = f(x)⋅g(x); f/g: X\{x ∈ X: g(x) = 0} → ℝ, x → (f/g)(x) = f(x)/g(x)

f: X → Y, g: Y → ℝ; compound function g∘f: X → ℝ, x → (g∘f)(x) = g(f(x)); the codomain of f must coincide with the domain of g

f: ℝ → ℝ, x → f(x) = cos(x); g: ℝ → ℝ, x → g(x) = x3; h: ℝ → ℝ, x → h(x) = ex; (g∘f)(x) = g(f(x)) = g(cos(x)) = (cos(x))3; (h∘g∘f)(x) = h(g(f(x))) = h(cos3(x)) = ecos3(x)

f: ℝ → ℝ, x → f(x) = 1-x2; g: [0,+∞) → ℝ, x → g(x) = √x; f(x) ∈ [0,+∞), f(X) must be in the domain of g ⇔ 1-x2 ≥ 0 ⇔ x ∈ [-1,1]; f: [-1,1] → ℝ, x → f(x) = 1-x2; g: [0,+∞] → ℝ, x → g(x) = √x; g∘f: [-1,1] → ℝ, x → g(f(x)) = √1-x2

The codomain of the internal function must be contained in the domain of the external function

f⋅g = g⋅f ⇔ f(x)g(x) = g(x)f(x), the product of two functions is commutative

f∘g ≠ g∘f, the composition product is not commutative

f(x) = |x|, x ∈ ℝ, f: ℝ → ℝ, x → f(x) = |x|; g(x) = sin(x), x ∈ ℝ, g: ℝ → ℝ, x → g(x) = sin(x); g∘f(x) = sin|x| → g∘f(3π/2) = sin|3π/2| = -1; f∘g(x) = |sin(x)| → f∘g(3π/2) = |sin(3π/2)| = |-1| = 1

f: A → B; 1) injective function if a1 ≠ a2 ⇒ f(a1) ≠ f(a2); 2) surjective function if ∀ b ∈ B, ∃ a ∈ A | f(a) = b; 3) bijective function if injective and surjective, that is ∀ b ∈ B ∃! a ∈ A | f(a) = b

An injective function, also known as injection, or one-to-one function, is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain

A surjective function, also known as onto, or a surjection, is a function f from a set X to a set Y, that for every element y in the codomain Y of f, there is at least one element x in the domain X of f such that f(x) = y; it is not required that x be unique; the function f may map one or more elements of X to the same element of Y

A bijective function, also known as bijection, or one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set; there are no unpaired elements; a bijective function f: X → Y is a injective and surjective mapping of a set X to a set Y

f: ℝ → ℝ, x → f(x) = x2; this function is non-injective and non-surjective; -1 = x2 has no solutions in ℝ, so -1 is not image of any x through f, it is non-surjective; 1 = x2 ⇔ x = ± 1, it is non-injective; f: ℝ → [0,+∞), x → f(x) = x2, it is surjective but non-injective; f: [0,+∞) → [0,+∞), x → f(x) = x2, it is bijective

Injective function ⇔ a line parallel to the x axis meets G(f) at most once

Surjective function ⇔ a line parallel to the x axis meets G(f) at least once

Bijective function ⇔ a line parallel to the x axis meets G(f) once and only once

An inverse function, or anti-function, is a function that reverses another function; if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x; g(y) = x if and only if f(x) = y; the inverse function of f is also denoted as f-1

f: X → Y, bijective, ∃! g: Y → X | {g(f(x)) = x; f(g(y)) = y}; the function g is the inverse function of f and it is written as f-1; the inverse function is the function that binds to y ∈ Y the only x ∈ X | f(x) = y ⇔ x = f-1(y)

f: [0,+∞) → [0,+∞), x → f(x) = x2; f-1: [0,+∞) → [0,+∞), x → f(x) = √x; x2 = y ⇔ x = √y

A function must be bijective to have an inverse function

(x,y) ∈ G(f) ⇔ y = f(x) ⇔ x = f-1(y) ⇔ (y,x) ∈ G(f-1); (x,y) and (y,x) are symmetric to the bisector of the 1° and 3° quadrant

G(f) and G(f-1) are symmetric to y = x

f-1(x) ≠ f(x)-1, inverse does not mean reciprocal; f(x)-1 = 1/f(x)

f: X → ℝ, 1) bounded from above if ∃ M ∈ ℝ | f(x) ≤ M ∀ x ∈ X, 2) bounded from belove if ∃ m ∈ ℝ | f(x) ≥ m ∀ x ∈ X, 3) bounded if ∃ m,M ∈ ℝ | m ≤ f(x) ≤ M ∀ x ∈ X

Symmetry: X | if x ∈ X ⇒ -x ∈ X and f: X → ℝ, 1) f is even if f(x) = f(-x) ∀ x ∈ X, 2) f is odd if f(x) = -f(-x) ∀ x ∈ X

f(x) = xn n ∈ ℕ; 1) if n is even, the function is even, x2 = (-x)2; 2) if n is odd, the function is odd, x3 = -(-x)3

If the function is even, the graph is symmetrical to the y axis; for example f(x) = x2 that is the quadratic function or parabola

If the function is odd, the graph is symmetrical to the origin; for example f(x) = x3 that is the cubic function

Monotonic function: f: X ⊆ ℝ → ℝ, 1) increasing if ∀ x1,x2 ∈ X | x1 ≤ x2 ⇒ f(x1) ≤ f(x2), 2) decreasing if ∀ x1,x2 ∈ X | x1 ≤ x2 ⇒ f(x1) ≥ f(x2)

f(x) = 1/x, it is a hyperbola; decreasing in (-∞,0) and decreasing in (0,+∞), but it is not decreasing in (-∞,0) ∪ (0,+∞); if it were decreasing ⇒ -1 = f(-1) ≥ f(1) = 1, but this is absurd

A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions, which repeat at intervals of 2π radians; any function that is not periodic is called aperiodic

f: X → ℝ is a periodic function if ∃ T ≠ 0 | f(x+T) = f(x) ∀ x ∈ X | x+T ∈ X; the smallest T > 0 is called period of the function

cos(x) = cos(x+2π) ∀ x ∈ ℝ, T = 2π; cos(x) = cos(x+4π) ∀ x ∈ ℝ

The fractional part (also known as mantissa) is a function that associates to each real number x its value minus its integer part, {x} = x-[x]; the mantissa function assumes all the values of the interval [0,1), it is periodic with a period equal to 1, it is neither even nor odd, and it is not an injective function, so it is not invertible

A polynomial is an expression consisting of variables, also called indeterminates, and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables

A polynomial is an elementary function

Polynomial: p(x) = anxn+...+a1x+a0 = nΣk=0 akxk, ai ∈ ℝ, an ≠ 0, n is the degree of the polynomial

A rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials

Rational function: f(x) = p(x)/q(x), p and q are polynomials; X = ℝ \ {x ∈ ℝ: q(x) = 0}; degree(f) = degree(p)-degree(q); for example f(x) = (x2+1)/(x3+3x+2), the degree is -1

Roots: xn = {bijective from [0,+∞) to [0,+∞), n even; bijective from ℝ to ℝ, n odd}; f-1(x) = nx, inverse function of f(x) = xn; nx = {defined from [0,+∞) to [0,+∞), n even; defined from ℝ to ℝ, n odd}

Real power and exponential: xn = x⋅x⋅...⋅x n-times; 34 = 3⋅3⋅3⋅3; 31/4 = 43; 2π = ?; π2 = ?; ar, a,r ∈ ℝ; 1) r ∈ ℕ ar = a⋅...⋅a r-times ∀ a ∈ ℝ; 2) r = p/q ∈ ℚ q ≠ 0 ar = ap/q = qap = (qa)p ∀ r ∈ ℚ if a > 0; r ∈ ℝ, r = p,α0α1α2... p ∈ ℤ, αi ∈ {0, 1, ..., 9}; 1.32 = 132/100; it is possible to construct a sequence of rational numbers that approximates r, considering a > 1 and r > 0, for a ∈ (0, 1) and r < 0 it is similar; rn = p.α0α1α2...αn, π = 3.141592, r0 = 3.1, r1 = 3.14, r2 = 3.141; 1) rn ∈ ℚ ∀ n, rn = (p.α0α1α2...αn)/10n; 2) {rn} is an increasing sequence; 3) {rn} is bounded, p ≤ rn < p+1; an increasing and bounded sequence is convergent; 4) limn→∞ rn = r, 0 ≤ r-rn = 0.00...0αn+1αn+2 ≤ 0.00...10000 = 1/10n → 0 ⇒ limn→∞ r-rn = 0 ⇔ limn→∞ rn = r; defining the sequence {arn}n∈ℕ; 1) the sequence is well defined because rn ∈ ℚ, considering that a > 0; 2) {arn}n∈ℕ is increasing because {rn}n is increasing and a > 1, if 0 < a < 1 it would be decreasing; 3) ap ≤ arn ≤ ap+1 because p ≤ r < p+1 and a > 1; considering the regularity theorem of monotonous sequences, if a sequence is monotone and increasing, then it admits a limit, so ∃ limn→∞ arn, and for definition ar := limn→∞ arn; if limn→∞ rn = limn→∞ sn = r, with {rn} and {sn} ⊆ ℚ, then ar = limn→∞ arn = limn→∞ asn; aras = ar+s, (ar)s = ars, valid for r,s ∈ ℚ but also for r,s ∈ ℝ

Real power: f(x) = xr ∀ x ∈ (0,+∞), r ∈ ℝ; if r > 0, f(x) = xr is defined for x = 0 with f(0) = 0; r ∈ ℕ, f(x) = xr is defined ∀ x ∈ ℝ

Exponential: f(x) = ax ∀ x ∈ ℝ, a > 0; f(x) = ex, ∀ x ∈ ℝ, exponential; f(x) = 3x, ∀ x ∈ ℝ, exponential of base 3; f(x) = πx, ∀ x ∈ ℝ, exponential of base π; ax, a > 1, is an increasing function, bounded below; ax, a ∈ (0,1), is a decreasing function, bounded below

Hyperbolic functions: cosh(x) = (ex+e-x)/2, ∀ x ∈ ℝ; sinh(x) = (ex-e-x)/2, ∀ x ∈ ℝ; tanh(x) = sinh(x)/cosh(x) = (ex-e-x)/(ex+e-x); they are similar to Euler's formulas: eix = cos(x)+i⋅sin(x), x ∈ ℂ, cos(x) = (eix+e-ix)/2, sin(x) = (eix-e-ix)/2i; cosh(x) is an even function and it is bounded below by the point x=0 y=1; sinh(x) is an odd function; tanh(x) is an odd function and it is bounded above by the point x=0 y=1 and bounded below by the point x=0 y=-1; the fundamental identity of hyperbolic functions is cosh2(x)-sinh2(x) = 1, that is similar to cos2(x)+sin2(x) = 1; the graph of the hyperbolic cosine function is called catenary

Trigonometric functions: a trigonometric function is a real function which relate an angle of a right-angled triangle to ratios of two side lengths; one radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle; the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle, that is θ = s/r, where θ is the subtended angle in radians, s is arc length, and r is radius; 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 120° = 2π/3, 135° = 3π/4, 150° = 5π/6, 180° = π, 210° = 7π/6, 225° = 5π/4, 240° = 4π/3, 270° = 3π/2, 300° = 5π/3, 315° = 7π/4, 330° = 11π/6, 360° = 2π; sin(x) is a periodic function with period of 2π, odd, bounded, -1 ≤ sin(x) ≤ 1 ∀ x ∈ ℝ; cos(x) is a periodic function with period of 2π, even, bounded, -1 ≤ cos(x) ≤ 1 ∀ x ∈ ℝ; tan(x) = sin(x)/cos(x) ∀ x ∈ ℝ \ {x ∈ ℝ | cos(x) = 0} ⇔ ∀ x ∈ ℝ \ {π/2 + kπ, k ∈ ℤ}, tan(x) is a periodic function with period of π, odd, not bounded; fundamental identity: cos2(x)+sin2(x) = 1