Algebra and Functions

• Algebra
• Functions
• see also:
  • Mensuration
  • Calculus
  • Circular functions
  • Coordinate geometry, parabolae, etc
  • Finance equations
  • Logarithmic, exponential and hyperbolic functions
  • Probability theory

Algebra:

• 11 field laws of algebra for real numbers (set of R):

  • field law:	addition	multiplication
    closure	a + b subset of R	ab subset of R
    commutative	a + b = b + a	ab = ba
    associative	a + (b + c) = (a + b) + c	a(bc) = (ab)c
    identity	a + 0 = a	a x 1 = a
    inverse	a + -a = 0	a x a-1 = 1, a <> 0
    distributive	a(b+c) =  ab + ac	as for additive

• some basic theorems of real numbers:
  • if a + c = b + c then a = b (additive cancellation law)
  • if ac = bc then a = b (multiplicative cancellation law)
  • if ab = 0, then either a = 0 or b = 0 or both equal zero (this helps us solve equations like (x-2)(x-3) = 0)
  • if a > b and b > c then a > c (transitive property)
  • if a > b then a + c > b + c (additive property)
  • if a > b then ac > bc (multiplicative property)
• some properties of surds:
  • closure law: when two different surds are added, they cannot be represented by another single surd
  • when unlike surds are multiplied, the product is a surd: sqrt(a) x sqrt(b) = sqrt (a x b)
  • however, when surds are multiplied or divided, the result may be a rational number, thus surds do not obey the closure laws
• polynomials:
  • a polynomial is an algebraic expression of the form:
    • P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a0, where n, n-1, ... are elements of J⁺; a_n, a_n-1, ... are elements of R, & x is a variable
  • remainder theorem:
    • enables one to find a remainder R without actual division
    • if P(x) is divided by (x-a) then the remainder R = P(a)
    • if P(a) = 0 ie. remainder is zero then (x-a) is a factor of P(x) - the factor theorem
  • quadratic polynomials:
    • ax² + bx + c where a <> 0, which can be expressed as a difference of two squares as long as the discriminant, b2 - 4ac >= 0 
• Some basic factors:
  • difference of two squares: x² - a² = (x - a)(x + a) 
  • x² + a² = no real factors
  • x³ - a³ = (x - a)(x² + ax + a²)
  • x³ + a³ = (x + a)(x² - ax + a²)
• Some basic expansions:
  • (x - a)² = x² - 2ax + a²
  • (x + a)² = x² + 2ax + a²
  • (x + a)³ = x³ + 3x²a + 3xa² + a³
  • (x - a)³ = x³ - 3x²a + 3xa² - a³
• Square root:
  • x² = y => x = +-sqrt(y), y >= 0;
• Modululus:
  • |x| = sqr x²
• Binomial theorem:
  • if n is set of N:
  • (x+a)ⁿ = xⁿ + nx^n-1a + n(n-1)x^n-2a²/2 +.....+ [n(n-1)(n-2)..(n-r+1)]x^n-ra^r / r! + .. + aⁿ
  •  
  • Linear approx. of (1 + x)ⁿ:
    • ~ 1 + nx, if x is small, n < R;
• Summation of a finite series:
  • Arithmetic:
    • a + (a + d) + ... + (a + (n - 1)d) = (n/2)(2a + (n - 1)d)
  • Geometric:
    • a + ar + ar2 + ... + ar^{(n - 1)} = a(1 - rⁿ)/(1 - r)
• Expansions:
  • sinx = x - (x³/3!) + (x⁵/5!) - (x⁷/7!) + ...;
  • cosx = 1 - (x²/2!) + (x⁴/4!) - (x⁶/6!) + ...;
  • ln(1+x) = x - (x²/2) + (x³/3) - (x⁴/4) + ...;
  • (1+x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! +.;

Functions:

• Cartesian Product:
  • eg. 2 sets:
    • X = {2,3}, Y = {4,5}.
    • X * Y = {(x,y): x E X, y E Y}
      • ie. {(2,4),(2,5),(3,4),(3,5)}
• Relation:
  • a set of ordered pairs: (domain,range)
• Ordered Pair Notation:
  • eg. {(x,y): y=x², xER }
• Mapping Notation:
  • eg. f: X -> Y where f(x)=x²
    • ie. X is mapped onto/into Y by the function f
    • X is the domain;
    • Y is the codomain;
    • f(x) is the range;
• Algebra of Functions:
  • (f+g)(x) = f(x) + g(x)
  • (fg)(x) = f(x) * g(x);
  • (f/g)(x) = f(x)/g(x), g(x) not= 0;
  • domain(D) = d(f+g) = d(fg) = d(f/g) = df intersection dg (NB. except for (f/g)(x) where exclude values x where g(x) = 0)
  • eg. if f(x) = x² and g(x) = 2x+4 then (f+g)(x) = x² + 2x + 4
• Composite Functions:
  • when x is operated on successively by f & then by g, ie. any occurrence of x in g(x) is substituted by f(x)
    • gof(x) = g[f(x)], domain = df,
    • range f must be subset domain g;
    • eg. if f(x) = x² and g(x) = 2x+4 then gof(x) = 2x² + 4
• Identity Functions:
  • foI = Iof = f, eg I(x)=x;
• Inverse Functions:
  • fof ^-1 = f ^-1 = I, r(f ^-1) = df, rf = d(f ^-1);
  • f must be a 1 to 1 function;