Exponential Functions

This page introduces the concept of exponential functions and their key properties. It covers the definition, algebraic rules, and behavior of these important mathematical functions.

Definition: An exponential function with base a is defined as the function that extends the geometric sequence (aⁿ) to non-integer exponents.

The page then delves into the algebraic properties of exponential functions, which are crucial for manipulating and solving exponential expressions.

Highlight: Key algebraic properties of exponential functions include:

• aˣ × aʸ = aˣ⁺ʸ
• aˣ ÷ aʸ = aˣ⁻ʸ
• (aˣ)ʸ = aˣʸ
• (ab)ˣ = aˣbˣ

Several examples are provided to illustrate these properties in action, helping students understand how to apply them in various scenarios.

Example: 7⁵·³ = 7¹⁵ and 2²ˣ⁺¹ = 2¹ × 2²ˣ = 2 × 4ˣ

The page also covers the behavior of exponential functions, particularly their direction of growth based on the base value.

Vocabulary: The terms "croissante" (increasing) and "décroissante" (decreasing) are used to describe the growth behavior of exponential functions.

Three specific examples are given to demonstrate how to determine whether an exponential function is increasing or decreasing:

1. f(x) = 2ˣ⁰·⁵ is decreasing because 0 < 0.5 < 1
2. g(x) = 0.1ˣ¹·⁸ is increasing because 1.8 > 1
3. h(x) = -7 × 0.3ˣ is decreasing because 0 < 0.3 < 1

These examples help reinforce the concept that exponential functions with bases between 0 and 1 are decreasing, while those with bases greater than 1 are increasing.

Highlight: The direction of growth for an exponential function depends on its base:

• If 0 < base < 1, the function is decreasing
• If base > 1, the function is increasing

This comprehensive overview provides students with a solid foundation in understanding exponential functions, their properties, and behavior, which is essential for more advanced mathematical concepts and real-world applications.