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Pre-Calculus Foundations

Number systems, equations, functions, and basic set theory - the prerequisites before diving into calculus, probability, and linear algebra.

Number Systems

N (naturals) subset Z (integers) subset Q (rationals) subset R (reals) subset C (complex)

  • Rationals: fractions p/q. Decimal terminates or repeats
  • Irrationals: sqrt(2), pi, e. Non-terminating, non-repeating
  • Fundamental theorem: every integer > 1 has unique prime factorization

Equations

Quadratic: ax^2 + bx + c = 0

  • Discriminant D = b^2 - 4ac
  • D > 0: two real roots, D = 0: one root, D < 0: no real roots
  • x = (-b +/- sqrt(D)) / (2a)
  • Vieta's: x1 + x2 = -b/a, x1 * x2 = c/a

Systems

  • Substitution: solve one for one variable, substitute
  • Elimination: add/subtract equations
  • Types: unique (intersecting), none (parallel), infinite (coincident)

Inequalities

  • When multiplying/dividing by negative: flip sign
  • Method of Intervals for rational: find zeros, check sign per interval
  • Quadratic: find roots, use parabola direction

Elementary Functions

Function Domain Key Property
Linear: y = kx + b R Constant rate of change
Quadratic: y = ax^2 + bx + c R Vertex at x = -b/(2a)
Exponential: y = a^x R Always positive, monotone
Logarithmic: y = log_a(x) x > 0 Inverse of exponential
Trigonometric: sin, cos R Period 2*pi, range [-1, 1]

Log properties: log(ab) = log(a) + log(b), log(a^n) = n*log(a), log_a(x) = ln(x)/ln(a)

Key identities: sin^2 + cos^2 = 1, e^(itheta) = cos(theta) + isin(theta)

Graph Transformations

  • f(x) + c: shift up
  • f(x - c): shift right
  • c*f(x): vertical stretch
  • f(cx): horizontal compression

Sets

  • Union A cup B: in A OR B
  • Intersection A cap B: in A AND B
  • Difference A \ B: in A but not B
  • De Morgan: (A cup B)^c = A^c cap B^c
  • Inclusion-exclusion: |A cup B| = |A| + |B| - |A cap B|
  • Power set: |P(A)| = 2^|A|

Combinatorics

  • Permutations (order matters): P(n,k) = n!/(n-k)!
  • Combinations (order doesn't): C(n,k) = n!/(k!(n-k)!)
  • Binomial theorem: (a+b)^n = sum C(n,k) * a^(n-k) * b^k
  • Pigeonhole principle: n items into m containers, n > m -> at least one has > 1

Sequences

  • Arithmetic: a_n = a_1 + (n-1)d, S_n = n(a_1+a_n)/2
  • Geometric: a_n = a_1 * r^(n-1), infinite sum (|r|<1): S = a_1/(1-r)
  • Compound interest: A = P(1+r)^n

See Also