General math 3rd quarter

• Function


  A rule that relates elements of a set of values (domain) to a set of second values (range) that follows a certain rule
• Domain


  The input
• Range


  The output
• Relation


  A set of ordered pairs
• Function


  • A relation where each element in the domain is related to only one element in the range by some rules
  • A set of ordered pairs such that no other pairs have the same first element but different second element
• A function's domain can be input that corresponds to only one point in the range
• f(x)


  40x if 0 ≤ x ≤ 10

  35x if 10 < x ≤ 30

  30x if x > 30
• Operation of Function


  (f + g)(x) = f(x) + g(x)

  (f - g)(x) = f(x) - g(x)

  (f * g)(x) = f(x) * g(x)

  (f / g)(x) = f(x) / g(x) where g(x) ≠ 0
• Composition of Function


  (f o g)(x) = f(g(x))

  (g o f)(x) = g(f(x))
• Rational Expression


  An expression that can be written as a ratio of two polynomials
• Rational Equation


  An equation involving rational expressions
• Rational Inequality


  An inequality involving rational expressions
• Rational Function


  A function in the form f(x) = P(x) / Q(x), where Q(x) ≠ 0
• One-to-One Function


  • For any x1, x2 in the domain of f, f(x1) = f(x2) implies x1 = x2
• One-to-One Functions in Real Life

  • Fingerprints
  • Passport
  • Social Media Passwords
  • DNA
  • Toothbrush
• Inverse Function


  If f is a one-to-one function with domain A and range B, then the inverse of f, denoted by f^(-1), is a function with domain B and range A
• Exponential Function


  A one-to-one function in the form y = b^x or f(x) = a * b^x, where a > 0, b > 0, b ≠ 1
• Exponential Equation


  y = (Initial Amount) * (Rate)^x

  Rate = (1 + %) if increasing (growth)

  Rate = (1 - %) if decreasing (decay)

  Rate = double, triple, quadruple... if growth

  Rate = half, third, etc... if decay
• Logarithmic Function


  The exponent that the base must be raised to produce a given number
• Logarithmic Form to Exponential Form

  • log_b x = y is equivalent to b^y = x