Section 2.1 - Functions of Intervals
Intervals - a set of points bound by two points on a line.
  • open interval (a, b) - all real numbers greater than 'a' and less than 'b' (a < x < b)
  • closed interval [c, d] - all real numbers greater than and equal to 'c' and less than and equal to 'd' (a ≤ x ≤ b)
  • half-open interval [m, n) - all real numbers greater than and equal to 'm' and less than 'n' (m ≤ x < n)
  • half-open interval (m, n] - all real numbers greater than 'm' and less than and equal to 'n' (m < x ≤ n)

Cannot have an interval that includes infinity.
  • [3, ∞] is not allowed because it includes infinity.
  • [-∞, 1) is not allowed because it includes infinity.
  • [-1, ∞) is allowed because it does not include infinity.

Functions
  • M = 1000(1 + 1%), M1 = 1000(1 + 2%), M2 = 1000(1 + 3%)
  • M = 1000 (1 + r) find interest on $1,000.
  • C = 5/9(F - 32) function to convert Fahrenheit to Celcius.
  • A = πr2 function to find area of a circle.

Definition - the relationship between x and y is a function if for any value of x, there exists one and only one value of y.

Examples of a Function
  • y = x
  • y = x2
  • x2 + y2 = 1 is not a function because more than one value of y exists for any one value of x (creates a circle)
A formula can be found to be a function simply by looking at the graph.

y = f(x)
  • y is a dependent variable
  • x is an independent variable
  • f is the function itself

Domains and Ranges
  • Domain of a function - is the set of all values of the independent variable for which the dependent variable is defined
    • y = 1/√x has a domain of x > 0
    • y = √x has a domain of x ≥ 0
  • Range of a function - is the set of all y values that can ocurr
    • y = x2 has a range of [0, ∞) because no negatives can exist with a "squared"
    • y = -√x has a range of (-∞, 0] because √x cannot be negative

Examples
  • y = √(x2 - 1)
    • Domain:
      • x2 - 1 ≥ 0
      • x2 ≥ 1
      • |x| ≥ 1
      • x ≥ 1 or x ≤ -1
    • Range: y ≥ 0 because square root cannot be negative

  • y = 1/(x-1)
    • Domain: all numbers but 1
    • Range: (-∞, ∞)

  • y = x/(x2 - 1)
    • Domain:
      • x2 - 1 ≠ 0
      • x2 ≠ 1
      • x ≠ ±1
    • Range: (-∞, ∞)

  • f(x) = x2 + 3x3
    • f(1) = 12 + 3(1)3
    • f(1) = 4

    • f(a2) = (a2)2 + 3(a2)3
    • f(a2) = a4 + 3a6

    • f(x - 2) = (x - 2)2 + 3(x - 2)3
    • f(x - 2) = x2 - 4x + 4 + 3(x3 - 3x22 + 3x24 - 23)
    • f(x - 2) = x2 - 4x + 4 + 3(x3 - 6x2 + 12x - 8)
    • f(x - 2) = 3x3 + (1 - 18)x2 + (-4 + 36)x - 20
    • f(x - 2) = 3x3 - 17x2 - 34x - 20

  • f(x) = √(x + 1), g(x) = x2 - 1
    • f(g(x)) = f(x2 - 1)
      • f(x2 - 1) = √(x2 - 1 + 1)
      • f(x2 - 1) = √(x2)
      • f(x2 - 1) = x

Homework for Section 2.1 - questions 3, 8, 12, 18, 19, 20, 29, 33, 34, 35