Understanding Functions and Their Evaluation

Aug 8, 2024

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.