Lecture Notes: Algebra of Functions

Key Concepts

• Objective: Understand how to find the sum, difference, product, and quotient of two functions.
• Functions Involved: Functions f and g.
• Domain: x should be in the domain of both functions.

Operations on Functions

1. Sum of Functions

   • Formula: ( (f + g)(x) = f(x) + g(x) )
   • Example: ( (f + g)(-2) = f(-2) + g(-2) = 8 + 3 = 11 )

2. Difference of Functions

   • Formula: ( (f - g)(x) = f(x) - g(x) )
   • Example: ( (f - g)(5) = f(5) - g(5) = -17 - 22 = -39 )

3. Product of Functions

   • Formula: ( (f \cdot g)(x) = f(x) \cdot g(x) )
   • Example: ( (f \cdot g)(0) = 3 \cdot (-3) = -9 )

4. Quotient of Functions

   • Formula: ( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} )
   • Example: ( \left( \frac{f}{g} \right)(2) = \frac{-5}{1} = -5 )

Application: Using a Graphing Calculator

• Steps:
  • Input f into y1 and g into y2.
  • Set table to "ask" mode.
  • Use the table to calculate function operations at specific x values.
  • Example: Verify calculations such as ( (f + g)(-2) = 12 ) using the calculator.

Graphical Interpretation

• Function Values as Y-values:
  • Example: ( (f + g)(8) = f(8) + g(8) = 3 + (-4) = -1 )
  • Analyze graph intersections and y-values for operations.

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