Notes on Lecture: Exponent Rules and Functions

Exponent Rules Introduction

• Exponent Definitions:
  • Example: 2^5 = 2 × 2 × 2 × 2 × 2 (5 times)
  • Base: the number being multiplied (2 in this case)
  • Exponent (or power): how many times the base is multiplied by itself (5 in this case)

Main Exponent Rules

Product Rule

• Rule: a^n × a^m = a^(n+m)
• Example: 2³ × 2⁴ = 2^(3+4) = 2⁷

Quotient Rule

• Rule: a^n ÷ a^m = a^(n-m)
• Example: 3⁶ ÷ 3² = 3^(6-2) = 3⁴

Power Rule

• Rule: (a^n)^m = a^(n*m)
• Example: (5⁴)³ = 5^(4×3) = 5¹²

Zero Exponent Rule

• Rule: a⁰ = 1 (for a ≠ 0)
• Justification: 2³ ÷ 2³ = 2^(3-3) = 2⁰ = 1

Negative Exponent Rule

• Rule: a^(-n) = 1/a^n
• Justification: 5⁷ × 5^(-7) = 5^(7-7) = 5⁰ = 1

Fractional Exponent Rule

• Rule: a^(1/n) = n√a
• Example: 64^(1/3) = ∛64 = 4

Distributing Exponents

• Over Product: (xy)^n = x^n * y^n
• Example: (5×7)³ = 5³ × 7³
• Over Quotient: (x/y)^n = x^n / y^n
• Example: (2/7)⁵ = 2⁵ / 7⁵
• Caution: Cannot distribute exponents over addition or subtraction.
  • Example: (a + b)ⁿ ≠ aⁿ + bⁿ

Summary of Exponent Rules

• Rules to Remember:
  • Product Rule
  • Quotient Rule
  • Power Rule
  • Zero Exponent Rule
  • Negative Exponent Rule
  • Fractional Exponent Rule
  • Distribution of Exponents

Examples of Simplifying Expressions

• Example 1: Simplifying using product and power rules.

Functions Introduction

• Definition: A function pairs each input with exactly one output.
• Notation: f(x) represents the output for input x.

Evaluating Functions

• Example:
  • f(2) = 2² + 1 = 5
  • f(5) = 5² + 1 = 26
• Complex Expressions:
  • f(a + 3) evaluates to (a + 3)² + 1 = a² + 6a + 10

Domains and Ranges

• Domain: All possible x-values.
• Range: All possible y-values.
• Example:
  • For f(x) = 1/x, domain x ≠ 0; range y ≠ 0

Inverses of Functions

• Definition: f⁻¹(x) reverses f(x) roles of x and y.
• Properties:
  • f(f⁻¹(x)) = x
  • f⁻¹(f(x)) = x
• Example of Non-Inverse: f(x) = x² (not one-to-one)

Graphs of Functions

• Key Functions:
  • Linear, Quadratic, Rational, Exponential, Logarithmic
• Transformations: Shifts, stretches, reflections.

Conclusion

• Keep rules and properties in mind when evaluating functions, working with logs, and understanding inverse functions.