Functions in Math

Study Notes

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)

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.

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.

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.

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.

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

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

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

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

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

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

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