Lecture Notes: Evaluating Functions and Composition of Functions

Key Topics

• Evaluating functions
• Composition of functions
• Function notation
• Domain of functions
• Difference quotient

Evaluating Functions

• Definition: Evaluating a function involves replacing the independent variable with another number or expression.
• Process: Replace the variable in the function with a blank space, plug in the required number/expression, and simplify.
• Example: For f(x) = -7x + 5, evaluating f(2) involves replacing x with 2 to get -7*2 + 5 = -9.
• Notation: f(2) vs. y = -7x + 5 – Function notation is preferred for clarity and descriptiveness.

Function Notation

• Purpose: To distinguish between different functions and make the evaluation process clear.
• Example: f(x) = -7x + 5 and g(x) = 3/2x - 1 are different functions despite using x as the variable.
• Benefit: Provides a clear input-output relationship.

Composition of Functions

• Definition: Composition involves evaluating functions where the input is another function or an expression.
• Process:
  1. Replace the variable with a blank space.
  2. Plug in the new expression.
  3. Simplify.
• Example: For h(x) = -2x^2 + x - 1, evaluating h(-1) involves replacing x with -1.

Identifying Non-Function Numbers

• Some numbers cannot be used as inputs because they result in undefined outputs (e.g., imaginary numbers, discontinuities).
• Example: Evaluating sqrt(x^2 - 3x) at x = 1 results in sqrt(1 - 3) = sqrt(-2), which is not a real number.

Domain of Functions

• Definition: The set of all possible input values for which the function is defined.
• Key Points:
  • Square roots require non-negative radicands.
  • Denominators cannot be zero.
• Example: For g(x) = (2x + 1)/(x - 5), x = 5 is outside the domain because it makes the denominator zero.

Difference Quotient

• Importance: Fundamental concept for calculus, particularly for understanding the slope of a curve at a point.
• Definition: The difference quotient is (f(x + h) - f(x)) / h.
• Process:
  1. Evaluate f(x + h) by replacing x with x + h.
  2. Simplify the expression.
  3. Combine f(x + h) and f(x) in the difference quotient formula.
  4. Simplify and cancel out h if possible.
• Example: For f(x) = x^2 + 1, f(x + h) = (x + h)^2 + 1 simplifies to x^2 + 2xh + h^2 + 1. The difference quotient then simplifies to 2x + h.

Practical Examples

• Evaluating with numbers: Plugging in specific numbers like 5, 0, or -1 to understand the output.
• Evaluating with expressions: Using expressions like 2x or x + 1 as inputs to test the function's flexibility.
• Domain issues: Identifying inputs that lead to undefined or non-real outputs, especially with square roots and denominators.

Summary

• Evaluating functions and understanding their domain is crucial for higher-level math, including calculus.
• Function notation and composition help in managing and describing functions clearly.
• The difference quotient is foundational for calculus, helping to understand the concept of derivatives.