Lecture Notes: Operations with Functions

Overview

• Objective: Understand operations with functions: addition, subtraction, multiplication, and division of two functions ( f(x) ) and ( g(x) ).
• Functions Given:
  • ( f(x) = x^2 - 3x )
  • ( g(x) = x + 6 )
• Operations:
  • ( (f+g)(x) )
  • ( (f-g)(x) )
  • ( (f \cdot g)(x) )
  • ( \frac{f}{g}(x) )
• Domain Considerations: Focus on intersections of domains and conditions for specific operations.

Function Addition: ( (f + g)(x) )

• Expression:
  • ( f(x) + g(x) = (x^2 - 3x) + (x + 6) )
  • Simplified to: ( x^2 - 2x + 6 )
• Domain:
  • All real numbers ((-\infty, \infty))
  • Both ( f(x) ) and ( g(x) ) have domains of all real numbers.

Function Subtraction: ( (f - g)(x) )

• Expression:
  • ( f(x) - g(x) = (x^2 - 3x) - (x + 6) )
  • Simplified to: ( x^2 - 4x - 6 )
• Domain:
  • All real numbers ((-\infty, \infty))
  • Intersection of domains for ( f(x) ) and ( g(x) ).

Function Multiplication: ( (f \cdot g)(x) )

• Expression:
  • ( (x^2 - 3x) \cdot (x + 6) )
  • Expand to: ( x^3 + 6x^2 - 3x^2 - 18x = x^3 + 3x^2 - 18x )
• Domain:
  • All real numbers ((-\infty, \infty))
  • Domains of both ( f(x) ) and ( g(x) ) are all real numbers.

Function Division: ( \frac{f}{g}(x) )

• Expression:
  • ( \frac{x^2 - 3x}{x + 6} )
  • Numerator can be factored but not simplified with the denominator.
• Domain:
  • All real numbers except ( x = -6 )
  • Exclude ( x + 6 = 0 ) (i.e., ( x = -6 )) to avoid division by zero.
  • Interval notation: ((-\infty, -6) \cup (-6, \infty))

Important Considerations

• Always check the domain restrictions when performing operations, especially with division, to avoid undefined expressions.
• Use interval notation to specify domains where there are exclusions.