Chapter 1
Linear Equations and Graphing
Course: Gre
Subject: Gre Quantitative Reasoning
Chapter: Foundations Of Functions And Composition
Introduction: When the Variable Is the Exponent
In previous lessons, we worked with expressions where the base contained the variable (like x²). Exponential functions reverse this relationship: the variable is in the exponent, and the base is a constant. These functions model growth and decay in the natural world—from population growth to radioactive decay to compound interest.
What Is an Exponential Function?
An exponential function has the form:
f(x) = a × b^x
where:
- a is the initial value (y-intercept, a ≠ 0)
- b is the base (b > 0, b ≠ 1)
- x is the exponent (independent variable)
Key Characteristics
| Feature | Description | | Variable location | Variable is in the exponent | | Base | Positive constant not equal to 1 | | Initial value | a is the y-intercept | | Domain | All real numbers | | Range | Positive real numbers (if a > 0) |