Definition and Core Properties
Exponential functions are mathematical expressions of the form f(x) = a * b^x, where a ≠ 0, b > 0, and b ≠ 1. Their key properties include a domain of all real numbers and a range of (0, ∞) for a > 0. The function is always positive, never crosses the x-axis, and has a horizontal asymptote at y = 0. The graph is strictly increasing if b > 1 or strictly decreasing if 0 < b < 1.
Key Mathematical Properties
Exponential functions exhibit unique algebraic properties, such as f(x + y) = f(x) * f(y), which underpins their multiplicative nature. They are one-to-one, ensuring an inverse (the logarithm function). The derivative is f'(x) = f(x) * ln(b), showing that the rate of change is proportional to the function's value, a hallmark of exponential growth or decay.
Practical Example
Consider f(x) = 2^x. At x = 0, f(0) = 1; at x = 1, f(1) = 2; at x = 2, f(2) = 4, illustrating rapid growth. For decay, g(x) = (1/2)^x starts at 1 and halves each time x increases by 1, approaching 0 asymptotically. These examples highlight how exponentials model real scenarios like population growth or radioactive decay.
Applications and Importance
Exponential functions are crucial in fields like finance (compound interest), biology (population dynamics), and physics (half-life calculations). Understanding their properties enables accurate modeling of rapid changes, distinguishing them from linear or polynomial functions, and is foundational for calculus and advanced mathematics.