Exponential functions are a fundamental concept in mathematics and have various real-world applications. An exponential function can be represented as f(x) = a * b^x, where the base 'b' is a positive constant and 'x' represents the exponent. The base 'b' determines the rate of growth or decay of the function.

Exponential functions exhibit unique characteristics. When the base 'b' is greater than 1, the function represents exponential growth. This means that as 'x' increases, the function value increases rapidly. On the other hand, when the base 'b' is between 0 and 1, the function represents exponential decay. In this case, as 'x' increases, the function value decreases exponentially.

Graphing exponential functions helps visualize their behavior. For exponential growth with a base greater than 1, the graph starts at the y-intercept (0, a) and rises steadily from left to right. Exponential decay, with a base between 0 and 1, starts at the y-intercept (0, a) and decreases exponentially from left to right, approaching the x-axis but never reaching it.

Exponential functions are used in various real-life scenarios. For example, population growth can be modeled using exponential functions. Compound interest, which is based on exponential growth, is another practical application. Understanding exponential functions equips you with the ability to solve real-world problems efficiently using mathematical tools.