Exponential Functions - Algebra-1

1. Fundamental Concepts

The core expression of an exponential function is y = a·bˣ, where:
- a represents the initial value (the value of the function when the independent variable x = 0)
- b is the base, which must satisfy b > 0 (a prerequisite for the exponential function to be meaningful)
- x is the independent variable, which can represent any real number

2. Key Concepts

The value of the base b directly determines the trend of the function:
- When 0 < b < 1, the function shows exponential decay (as x increases, the value of y gradually decreases)
- When b > 1, the function shows exponential growth (as x increases, the value of y increases rapidly)

3. Examples

1. Given the exponential function y = 5·bˣ, and y = 20 when x = 2, find the value of b (where b > 0).
Solution: Substitute x = 2 and y = 20 into the function, we get 20 = 5·b², that is, b² = 4. Since b > 0, b = 2.

2. Given the exponential function y = a·(3/2)ˣ, and y = 4/3 when x = -1, find the value of the function when x = 3.
Solution: First, find a. Substitute x = -1 and y = 4/3 into the function, we get 4/3 = a·(3/2)⁻¹ = a·(2/3), so a = 2.
Then, find the value of y when x = 3: y = 2·(3/2)³ = 2·(27/8) = 27/4.

3. Problem: Solve for $$x$$ in the equation $$3^x = 27$$

Step-by-Step Solution:

Recognize that $$27 = 3^3$$
Set the exponents equal since the bases are the same: $$x = 3$$

Validation: Substitute $$x=3$$ → Original: $$3^3 = 27$$; Simplified: $$27 = 27$$ ✓

4. Problem-Solving Techniques

Identify the function form: Determine whether a function is an exponential function based on the expression y = a·bˣ, with a focus on checking if the base b satisfies b > 0.
Analyze the nature of the base: Directly judge whether the function is growing or decaying based on the relationship between b and 1 (b > 1 or 0 < b < 1).
Calculate the function value: When the value of the independent variable x is given, directly substitute it into the expression and calculate the corresponding y value using the rules of exponential operations (such as bⁿ·bᵐ = bⁿ⁺ᵐ, (bⁿ)ᵐ = bⁿᵐ, etc.).
Solve for parameters a or b: If the values of x and the corresponding y are known, the initial value a or the base b can be solved through equations (pay attention to the restriction that b > 0).