Algebra of Functions and Function Composition

Algebra of Functions and Function Composition

• The sum, difference, product, and quotient of functions are defined for all x common to the domains of both functions.
• Sum: (f + g)(x) = f(x) + g(x)
• Difference: (f – g)(x) = f(x) – g(x)
• Product: (fg)(x) = (f(x))(g(x))
• Quotient: (f / g)(x) = f(x) / g(x), provided g(x) ≠ 0

Examples of Function Operations

• Given f(x) = x² + 1 and g(x) = 3x – 5:

  • (f + g)(x) = x² + 3x – 4, domain: (-∞, ∞)
  • (f – g)(x) = x² – 3x + 6, domain: (-∞, ∞)
  • (fg)(x) = 3x³ – 5x² + 3x – 5, domain: (-∞, ∞)
  • (f / g)(x) = (x² + 1) / (3x – 5), domain: (-∞, 5/3) U (5/3, ∞)
  • (f + g)(3) = 14
  • (fg)(4) = 119

• Given f(x) = x² – 5 and g(x) = √(6 – x)

  • (fg)(x) = x²√(6 – x) – 5√(6 – x), domain: (-∞, 6]

• Given m(x) = (x – 4) / (x² – 25) and n(x) = (x – 4) / (5 – x)

  • (m / n)(x) = -1 / (x + 5), x ≠ -5, 5, 4, domain: (-∞, -5) U (-5, 4) U (4, 5) U (5, ∞)
  • (m + n)(x) = (x³ – x² – 16x – 20) / ((x² – 25)(5 – x)), domain: (-∞, -5) U (-5, 5) U (5, ∞)

Difference Quotient

• The difference quotient for a function f(x) is given by:
  • ( f(x + h) – f(x) ) / h
• This expression determines the average rate of change of the function f(x) over an interval of length h.

Examples of Difference Quotient

• Given f(x) = 4x + 7:

  • f(x + h) – f(x) / h = 4, h ≠ 0

• Given f(x) = 2x² – 3x + 4:

  • f(x + h) – f(x) / h = 4x + 2h – 3, h ≠ 0