algebra

• Algebraic Expression


  A combination of numbers and letters connected by arithmetic operations
• Constant


  A number that does not change
• Variable


  A symbol, usually a letter, that changes value depending on the problem
• Like Terms


  Terms having the same variables wherein addition and subtraction are possible
• Unlike Terms


  Terms having different variables wherein addition and subtraction are impossible
• Polynomial


  An algebraic expression in the form ax^n wherein "a" is a real number and "n" is an integer that can be characterized based on the number of terms (e.g. monomial, binomial) or based on the order (highest-degree term: e.g. first, second, third)
• FOIL Method


  A technique used for multiplying binomials; stands for "First, Outside, Inside, Last"
• Evaluation


  A process by which a variable is replaced with a number such that arithmetic operations can be performed
• Laws on Exponents


  • Addition/Subtraction
  • Multiplication
  • Division
  • Power of a Power
  • Power of a Product
  • Power of a Fraction
  • Zero Exponent
  • Negative Exponent
  • Fractional Exponent
• Equation


  A statement in mathematics that shows equality between two mathematical expressions, separated by an equals sign, =
• Function


  Relates two sets of variables, where each object in the first set (domain) matches with exactly one in the second set (range). A function is often denoted by a symbol f(x) (pronounced f of x), where x is the element of the domain, and f(x) is the element of the range
• Vertical Line Test


  • If a vertical line crosses the curve only once, it's a function. Otherwise, not a function
• Domain


  Possible inputs for a function (x)
• Range


  All possible outputs of a function f(x)
• Types of Functions


  • Linear Function (x^1)
  • Quadratic Function (x^2)
  • Higher-order Polynomials (x^n, where n is an integer > 2)
  • Rational Function
• Equation of a Line Forms

  • Slope-Intercept
  • Point-Slope
  • Standard Form
  • Intercept Form
  • Vertical
  • Horizontal
• Quadratic Equation Forms


  • Standard Form
  • Point-Slope
  • Intercept Form
• Properties of Rational Functions

  • Domain
  • Range
  • Holes
  • Vertical Asymptote
  • Horizontal Asymptote
  • Oblique or Slant Asymptote
• Inverse Function


  The inverse of a function, f(x) is represented f^(-1)(y)
• Composite Function


  Functions within a function f[g(x)] or (f ∘ g)(x)
• Piecewise Function


  A function that is defined on a sequence of intervals
• N


  One greater than the highest power of D
• Logarithmic Function


  Inverse of an exponential function, expressed as x is the logarithm of n to the base b, aˣ = n, x = logₐn
• Types of Logarithm


  • Common Logarithm - logarithms with base 10
  • Natural Logarithm - logarithm with base e = 2.7182828
• Logarithmic Laws


  • Products: logₐ mn = logₐm + logₐn
  • Ratios: logₐ m/n = logₐm - logₐn
  • Powers: logₐ nᵖ = p logₐn
  • Roots: logₐ ˢ√n = 1/s logₐn
  • Change of bases: logₐn = logₜn / logₜa
• Inequalities


  Mathematical statement that compares algebraic expressions using greater than (>), less than (<), and other inequality symbols
• To solve for Inequalities


  Solve similar to standard equations, but note that multiplying or dividing values with negative signs reverses the inequality (i.e. > will be <, and vice versa)
• Systems of Equations


  Solving two or more equations to provide a common solution or value of variables (such as in terms of x & y)
• Solving by Elimination


  Set up equations similar to normal addition/subtraction that allow one variable to be canceled, solve for the remaining equation of the variable, substitute the value of the variable to one of the original equations to solve for the other value.
• Solving by Substitution


  Setup one equation in terms of x or y, substitute variable of the setup equation with the other equation, solve for missing variable and substitute value to the setup equation to solve for the other value.
• Set


  A well-defined collection of distinct mathematical objects called elements, usually named using capital letters, elements are enclosed in braces, unordered, meaning the order or arrangement of the elements does not matter
• Set Definition Types


  • Statement/Description: describes the elements of a set in words/well-defined terms
  • Roster: lists all of the elements explicitly
  • Set Builder: has a general form {x | properties}
• Set Theory Symbols


  • { }: Set
  • ∪: Union
  • ∩: Intersection
  • ⊆: Subset
  • ⊄: Not Subset
  • ⊂: Proper/Strict Subset
  • ⊃: Proper/Strict Superset
  • ⊇: Superset
  • Ø: Empty set
  • P(C): Power Set
  • ⊅: Not Superset
  • = : Equality
  • \ or -: Relative Component
  • Aᶜ: Complement
  • ∆: Symmetric Difference
  • ∈: Element of
  • (a, b): Ordered Pair
  • ∉: Not element of
  • |B|: Cardinality
  • ×: Cartesian Product
  • N₁: Natural/Whole numbers set (without zero)
  • N₀: Natural/Whole numbers set (with zero)
  • Q: Rational Numbers Set
  • Z: Integer Numbers Set
  • C: Complex Numbers Set
  • R: Real Numbers Set
• Set Operations


  • Union
  • Intersection
  • Complement
  • Difference