Functions in Math

Functions

• Domain and Range: Domain is the set of input values, and range is the set of output values.
• Types of Functions:
  • One-to-One: Each output value corresponds to exactly one input value.
  • Onto: Every output value is reached by at least one input value.
  • Linear: f(x) = mx + b, where m is the slope and b is the y-intercept.
  • Quadratic: f(x) = ax^2 + bx + c, where a, b, and c are constants.
• Function Operations:
  • Composition: (f ∘ g)(x) = f(g(x))
  • Addition and Subtraction: (f + g)(x) = f(x) + g(x), (f - g)(x) = f(x) - g(x)
  • Multiplication and Division: (f × g)(x) = f(x) × g(x), (f ÷ g)(x) = f(x) ÷ g(x)

Analytic Geometry

• Coordinate Systems:
  • Cartesian Coordinates: (x, y) represents a point in the plane.
  • Polar Coordinates: (r, θ) represents a point in the plane, where r is the distance from the origin and θ is the angle from the x-axis.
• Equations of Lines:
  • Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
  • Point-Slope Form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
  • Standard Form: Ax + By = C, where A, B, and C are constants.
• Equations of Circles:
  • Standard Form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.

Trigonometry

• Angles and Triangles:
  • Degrees and Radians: 1 radian = 180/π degrees.
  • Right Triangles: angles, side lengths, and trigonometric functions (sine, cosine, and tangent).
• Trigonometric Functions:
  • Sine: sin(θ) = opposite side / hypotenuse.
  • Cosine: cos(θ) = adjacent side / hypotenuse.
  • Tangent: tan(θ) = opposite side / adjacent side.
• Identities and Formulas:
  • Pythagorean Identity: sin^2(θ) + cos^2(θ) = 1.
  • Sum and Difference Formulas: sin(a + b) = sin(a)cos(b) + cos(a)sin(b), etc.

Differentiation

• Limits:
  • Definition of a Limit: lim x→a f(x) = L if f(x) approaches L as x approaches a.
  • Properties of Limits: sum, product, and chain rule.
• Derivatives:
  • Definition of a Derivative: f'(x) = lim h→0 [f(x + h) - f(x)]/h.
  • Rules of Differentiation:
    • Power Rule: if f(x) = x^n, then f'(x) = nx^(n-1).
    • Product Rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
    • Quotient Rule: if f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2.

Integration

• Definite Integrals:
  • Definition of a Definite Integral: ∫[a,b] f(x)dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
  • Properties of Definite Integrals: linearity, additivity, and substitution.
• Indefinite Integrals:
  • Definition of an Indefinite Integral: ∫f(x)dx = F(x) + C, where F(x) is the antiderivative of f(x) and C is the constant of integration.
  • Basic Integration Rules:
    • Power Rule: ∫x^n dx = (x^(n+1))/(n+1) + C.
    • Substitution Method: ∫f(u) du = F(u) + C, where u = u(x) and du/dx = dx/dx.
    • Integration by Parts: ∫u(x)v'(x) dx = u(x)v(x) - ∫v(x)u'(x) dx.

Functions

• Domain: set of input values, and range: set of output values

Types of Functions

• One-to-One: each output value corresponds to exactly one input value
• Onto: every output value is reached by at least one input value
• Linear: f(x) = mx + b, where m is the slope and b is the y-intercept
• Quadratic: f(x) = ax^2 + bx + c, where a, b, and c are constants

Function Operations

• Composition: (f ∘ g)(x) = f(g(x))
• Addition and Subtraction: (f + g)(x) = f(x) + g(x), (f - g)(x) = f(x) - g(x)
• Multiplication and Division: (f × g)(x) = f(x) × g(x), (f ÷ g)(x) = f(x) ÷ g(x)

Analytic Geometry

Coordinate Systems

• Cartesian Coordinates: (x, y) represents a point in the plane
• Polar Coordinates: (r, θ) represents a point in the plane, where r is the distance from the origin and θ is the angle from the x-axis

Equations of Lines

• Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept
• Point-Slope Form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope
• Standard Form: Ax + By = C, where A, B, and C are constants

Equations of Circles

• Standard Form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius

Trigonometry

Angles and Triangles

• Degrees and Radians: 1 radian = 180/π degrees
• Right Triangles: angles, side lengths, and trigonometric functions (sine, cosine, and tangent)

Trigonometric Functions

• Sine: sin(θ) = opposite side / hypotenuse
• Cosine: cos(θ) = adjacent side / hypotenuse
• Tangent: tan(θ) = opposite side / adjacent side

Identities and Formulas

• Pythagorean Identity: sin^2(θ) + cos^2(θ) = 1
• Sum and Difference Formulas: sin(a + b) = sin(a)cos(b) + cos(a)sin(b), etc.

Differentiation

Limits

• Definition of a Limit: lim x→a f(x) = L if f(x) approaches L as x approaches a
• Properties of Limits: sum, product, and chain rule

Derivatives

• Definition of a Derivative: f'(x) = lim h→0 [f(x + h) - f(x)]/h
• Rules of Differentiation:
  • Power Rule: if f(x) = x^n, then f'(x) = nx^(n-1)
  • Product Rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
  • Quotient Rule: if f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2

Integration

Definite Integrals

• Definition of a Definite Integral: ∫[a,b] f(x)dx = F(b) - F(a), where F(x) is the antiderivative of f(x)
• Properties of Definite Integrals: linearity, additivity, and substitution

Indefinite Integrals

• Definition of an Indefinite Integral: ∫f(x)dx = F(x) + C, where F(x) is the antiderivative of f(x) and C is the constant of integration
• Basic Integration Rules:
  • Power Rule: ∫x^n dx = (x^(n+1))/(n+1) + C
  • Substitution Method: ∫f(u) du = F(u) + C, where u = u(x) and du/dx = dx/dx
  • Integration by Parts: ∫u(x)v'(x) dx = u(x)v(x) - ∫v(x)u'(x) dx